On the structure of the $C^*$-algebra generated by the field operators and spectral analysis of the operators affiliated to it

TL;DR

This paper analyzes the structure of a specific $C^*$-algebra generated by field operators, revealing its grading by subspaces, and describes the spectral properties of operators affiliated to it, akin to N-body Hamiltonians.

Contribution

It provides a detailed structural and spectral analysis of the $C^*$-algebra generated by field operators, including a grading by subspaces and an HVZ-type theorem for spectral characterization.

Findings

The $C^*$-algebra is graded by finite dimensional subspaces of the symplectic space.

Self-adjoint operators affiliated to the algebra exhibit a many-channel structure similar to N-body Hamiltonians.

The essential spectrum of these operators is characterized by an HVZ-type theorem.

Abstract

We show that the $C^*$-algebra generated by the field operators associated to a symplectic space $\Xi$ is graded by the semilattice of all finite dimensional subspaces of $\Xi$. If $\Xi$ is finite dimensional we give a simple intrinsic description of the components of the grading, we show that the self-adjoint operators affiliated to the algebra have a many channel structure similar to that of N-body Hamiltonians, in particular their essential spectrum is described by a kind of HVZ theorem, and we point out a large class of operators affiliated to the algebra.

Equations

$${\sum }_{i\in I}^{\mathrm{c}}{\mathcal{A}}_i\doteq \text{ norm closure of the linear sum }{\sum }_{i\in I}{\mathcal{A}}_i.$$

$${\mathcal{A}}_1{\mathcal{A}}_2\dots {\mathcal{A}}_n\doteq \text{ linear span of the products }{A}_1{A}_2\dots A_n\text{ with }{A}_i\in {\mathcal{A}}_i$$

$${\mathcal{A}}_1\cdot {\mathcal{A}}_2\cdot \dots \cdot {\mathcal{A}}_n\doteq \text{ closure of }{\mathcal{A}}_1{\mathcal{A}}_2\dots {\mathcal{A}}_n.$$

$$\theta (A)\in \mathcal{C}\forall \theta \in C_0(\mathbb{R})\iff (A-z{)}^{-1}\in \mathcal{C}\text{ for some }z\notin \sigma (A).$$

$$A\tilde{\in }\mathcal{C}\iff A\in \mathcal{C}\text{ or }A\text{ is an observable affiliated to }\mathcal{C}.$$

$$\mathcal{C}(E)\mathcal{C}(F)\subset \mathcal{C}(E+F)\forall E,F\in \mathcal{S}.$$

$$C_0(X/Y)\cdot C_0(X/Z)=C_0(X/(Y\cap Z))\forall Y,Z\in \mathbb{G}(X).$$

$${\mathcal{G}}_X\doteq \text{ closure of }{\mathring{\mathcal{G}}}_X={\sum }_{Y\in \mathbb{G}(X)}^{\mathrm{c}}{C}_0(X/Y)\subset C_{\mathrm{b}}^{\mathrm{u}}(X)$$

$$C_0(X/Y)=C^{\ast }({\phi }_1)\cdot C^{\ast }({\phi }_1)\cdot \dots \cdot C^{\ast }({\phi }_n).$$

$$E\in \mathbb{G}(\Xi )\Rightarrow \exists F\in {\mathbb{G}}_{\mathrm{s}}(\Xi )\text{ such that }E\subset F.$$

$$W(\xi +\eta )={\mathrm{e}}^{\frac{\mathrm{i}}{2}\sigma (\xi ,\eta )}W(\xi )W(\eta )\forall \xi ,\eta \in \Xi \text{and}\underset{t\to 0}{w-lim}W(t\xi )=1(t\in \mathbb{R}).$$

$$\underset{\xi \in E,\xi \to 0}{\lim }\Vert W(\xi {)}^{\ast }TW(\xi )-T\Vert =0\text{for all }T\in \mathcal{K}\text{ and }E\in \mathbb{G}(\Xi ).$$

$$\mathcal{F}$$

$$=\text{C∗-algebra generated by the operators }{R}_{\xi }(z)\text{ with }\xi \in \Xi .$$

$$\mathcal{F}(E)$$

$$=C^{\ast }(\varphi ({\xi }_1))\cdot C^{\ast }(\varphi ({\xi }_2))\cdot \dots \cdot C^{\ast }(\varphi ({\xi }_n))$$

$$=\text{ norm closed linear span of products }{R}_{{\xi }_1}(\mathrm{i}{\lambda }_1)\dots R_{{\xi }_n}(\mathrm{i}{\lambda }_n)$$

$$\text{ with }{\lambda }_1,\dots ,{\lambda }_n\in \mathbb{R}\setminus \{0\}.$$

$$E,F\in \mathbb{G}(\Xi )\Rightarrow \mathcal{F}(E)\cdot \mathcal{F}(F)=\mathcal{F}(E+F),$$

$$\mathring{\mathcal{F}}\doteq {\sum }_{E\in \mathbb{G}(\Xi )}\mathcal{F}(E)\text{ is a linear direct sum and is dense in }\mathcal{F},$$

$$\text{if }\mathcal{S}\subset \mathbb{G}(\Xi )\text{ is finite then }{\sum }_{E\in \mathcal{S}}\mathcal{F}(E)\text{ is norm closed}.$$

$$\mathcal{F}={\sum }_{E\in \mathbb{G}(\Xi )}^{\mathrm{c}}\mathcal{F}(E).$$

$$\mathring{\mathcal{F}}(\mathcal{S})\doteq {\sum }_{E\in \mathcal{S}}\mathcal{F}(E)\text{and}\mathcal{F}(\mathcal{S})\doteq {\sum }_{E\in \mathcal{S}}^{\mathrm{c}}\mathcal{F}(E).$$

$${\mathcal{F}}_E\doteq {\sum }_{F\subset E}^{\mathrm{c}}\mathcal{F}(F),{\mathcal{F}}_E^{\prime }\doteq {\sum }_{F\not\subset E}^{\mathrm{c}}\mathcal{F}(F),{\mathcal{F}}_{\supset E}\doteq {\sum }_{F\supset E}^{\mathrm{c}}\mathcal{F}(F).$$

$${\mathcal{F}}_E=C^{\ast }(\varphi (\xi )\mid \xi \in E).$$

$$\mathcal{F}={\mathcal{F}}_E+{\mathcal{F}}_E^{\prime }\text{ and }{\mathcal{F}}_E\cap {\mathcal{F}}_E^{\prime }=0.$$

$$T={\sum }_{F}T(F)\in \mathring{\mathcal{F}}\Rightarrow {\mathcal{P}}_{E}T={\sum }_{F\subset E}T(F).$$

$${\mathcal{F}}_{E\cap F}={\mathcal{F}}_E\cap {\mathcal{F}}_F\text{and}{\mathcal{P}}_{E\cap F}={\mathcal{P}}_E{\mathcal{P}}_F={\mathcal{P}}_F{\mathcal{P}}_E.$$

$$\underset{\xi \to 0}{\lim }\Vert [W(\xi ),T]\Vert =0,\underset{\xi \in E,\xi \to 0}{\lim }\Vert (W(\xi )-1)T\Vert =0,[W(\xi ),T]=0\forall \xi \in E^{\sigma }.$$

$$\xi \mapsto W(\xi )TW(\xi {)}^{\ast }\text{ is norm continuous on finite dimensional subspaces of }\Xi .$$

$$\mathcal{E}\doteq {\sum }_{E\in \mathbb{G}(\Xi )}^{\mathrm{c}}\mathcal{E}(E)$$

$$\Phi (\mathcal{O})\doteq C^{\ast }(\varphi (\xi )\Gamma (A)\mid \xi \in \Xi ,A\in \mathcal{O},\Vert A\Vert <1)$$

$$E\subset F\Rightarrow \mathrm{Sp}({\mathcal{P}}_{E}T)\subset \mathrm{Sp}({\mathcal{P}}_{F}T)\forall T\in \mathcal{F}.$$

$$\mathrm{S}{\mathrm{p}}_{\mathrm{ess}}(T)={\bigcup }_{H\in \mathbb{H}(\Xi )}\mathrm{Sp}({\mathcal{P}}_{H}T)\forall T\tilde{\in }\mathcal{F}.$$

Full text

AMS Subject classification (2010): 46L60 (Primary), 47A10, 47L90, 81Q10, 81S05, 81S30 (Secondary).

Key words and phrases: canonical commutation relations, quantum field theory, $C^{*}$-algebra, spectral theory, Keyessential spectrum, HVZ theorem.

1. 1 Introduction

2. 1.1

3. 1.2 Grassmann C*-algebra

4. 1.3

5. 1.4

6. 1.5

7. 1.6

8. 1.7

9. 1.8

10. 2 Field algebra

11. 2.1 Kastler algebra

12. 2.1.1 Measure spaces

13. 2.1.2 Definition

14. 2.1.3 Representations

15. 2.1.4 C*-norm on $L^{1}$

16. 2.2 Components of the field algebra

17. 2.3 Field algebra

18. 2.4 Projection morphisms

19. 2.5 Field observables

20. 2.5.1 Fourier algebra and observables

21. 2.5.2 Field observables

22. 2.6 Field algebra in a representation

23. 3 Finite dimensional symplectic spaces

24. 3.1 Intrinsic description of $\mathscr{F}(E)$

25. 3.1.1 ​​​

26. 3.1.2 ​​​

27. 3.1.3 ​​​

28. 3.1.4 ​​​

29. 3.1.5 ​​​

30. 3.2 The ideal $\mathscr{F}(\Xi)$

31. 3.3 HVZ theorem

32. 4 Subalgebras of $\mathscr{F}$ and Hamiltonians affiliated to them

33. 4.1 Half-Lagrangian decompositions and magnetic fields

34. 4.2 Subalgebras of $\mathscr{F}$

35. 4.3 N-body Hamiltonians and beyond

36. 5 Extended field C*-algebra

37. A C*-algebras graded by Grassmannians

38. A.1 Main facts

39. A.2 Quotient algebra

40. A.3 Essential spectrum

41. A.4 Affiliation criteria

1. Introduction

The field operator algebra of a symplectic space $\Xi$ was introduced by D. Kastler as “the object of main interest for field theory” in [40, §6]. Kastler constructs a $C^{*}$-algebra $\mathscr{M}$ involving the twisted convolution product and the field $C^{*}$-algebra $\mathscr{L}$ is mentioned as a subalgebra of interest only at the end of his paper (he devotes just over a page to this issue).

Independently of Kastler’s work, D. Buchholz and H. Grundling introduced the resolvent CCR algebra in [9, 10] as an alternative of the Weyl CCR algebra with more convenient properties [12]. Loosely speaking, if we work in a representation of $\Xi$ and denote $\phi(\xi)$ the field operator at the point $\xi\in\Xi$, then the Weyl algebra is the $C^{*}$-algebra generated by the exponentials $\mathrm{e}^{\mathrm{i}\phi(\xi)}$ while the resolvent algebra is the $C^{*}$-algebra generated by the resolvents $R_{\xi}=(\mathrm{i}-\phi(\xi))^{-1}$, where $\xi$ runs over $\Xi$. On a more abstract level, the resolvent CCR algebra is the universal $C^{*}$-algebra generated by a family of operators $\{R_{\xi}\}_{\xi\in\Xi}$ satisfying certain relations suggested by the preceding definition of these operators in a representation, cf. [10, Defs. 3.1 and 3.4]. In fact the resolvent and field algebra are canonically isomorphic as follows from [40, Th. 23]. The resolvent algebra has been studied and used in the treatment of quantum systems with a finite or infinite number of degrees of freedom in a series of papers by D. Buchholz [13, 14, 15, 16] and D. Bahns and D. Buchholz [3].

If $\Xi$ is finite dimensional a certain class of $C^{*}$-subalgebras of $\mathscr{M}$ has been pointed out in [7]: for each finite stable under sums set $\mathcal{S}$ of subspaces of $\Xi$ a $C^{*}$-subalgebra $\mathscr{L}(\mathcal{S})\subset\mathscr{M}$ is defined which has a structure similar to the $C^{*}$-algebra generated by the Hamiltonians of an $N$-body system. The self-adjoint operators affiliated to such subalgebras generalize the usual $N$-body Hamiltonians and have a similar many channel structure and spectral theory. In [29] the theory is extended to infinite sets $\mathcal{S}$, in particular a $C^{*}$-algebra $\mathscr{L}$, called (graded) symplectic $C^{*}$-algebra of $\Xi$, is associated to the set of all subspaces of $\Xi$. Only after reading the paper [10] by D. Buchholz and H. Grundling did we realise that $\mathscr{L}$ is just Kastler’s field algebra, or resolvent CCR algebra. The extension to infinite dimensional $\Xi$ of our construction of the symplectic $C^{*}$-algebra is easy. Thus “field algebra”, “resolvent CCR algebra”, and “(graded) symplectic algebra” should be considered as synonyms; we mainly use the terminology “field algebra” for historical reasons.

We make a preliminary comment on our notations. Five $C^{*}$-algebras (and their subalgebras) appear in our arguments: $\mathscr{M},\mathscr{L}$, $\mathscr{K},\mathscr{F}$, $\mathscr{E}$. But in fact $\mathscr{M}$ is canonically isomorphic to $\mathscr{K}$ and $\mathscr{L}$ is canonically isomorphic to $\mathscr{F}$. More precisely, $\mathscr{M}$ and $\mathscr{L}$ are the “abstract” Kastler and field algebra (defined in (2.6) and (2.18)) and involve the twisted convolution product associated to symplectic space $\Xi$. The $C^{*}$-algebras $\mathscr{K},\mathscr{F}$ are images of $\mathscr{M},\mathscr{L}$ constructed via a (continuous) representation $W$ of $\Xi$, see Definition 1.4 (these algebras depend on $\Xi$ and $W$ but we do not indicate it in notations unless really needed). The algebra $\mathscr{E}$, called extended field $C^{*}$-algebra, is defined only in representations of $\Xi$, it contains $\mathscr{F}$, and is needed in the spectral theory of particles with nonzero spin, e.g. N-body Dirac and Pauli Hamiltonians (see §1.4 for a short presentation and Sec. 5 for a detailed study).

Our results on the spectral analysis of the operators affiliated to $\mathscr{F}$ require $\mathscr{F}\cap K(\mathcal{H})\neq 0$, where $\mathcal{H}$ is the Hilbert space on which is represented $\mathscr{F}$ and $K(\mathcal{H})$ is the algebra of compact operators on $\mathcal{H}$. Indeed, a description of the quotient $C^{*}$-algebra $\mathscr{F}/K(\mathcal{H})$ allows one to compute the essential spectrum of the observables affiliated to $\mathscr{F}$ and to prove the Mourre estimate with respect to convenient conjugate operators. But $\mathscr{F}\cap K(\mathcal{H})=0$ if $\dim\Xi=\infty$, hence the case of finite dimensional symplectic spaces is important for us.

A summary of our more significative results concerning the field algebra $\mathscr{F}$ may be found in the subsections 1.3–1.8 of this introduction, while in subsections 1.1 are 1.2 we recall or introduce the main notions and notations. We mention that we are forced to use a more general notion of observable instead of self-adjoint operator because in general images through morphisms and strong limits of self-adjoint operators affiliated to a $C^{*}$-algebra are not self-adjoint operators in the usual sense (e.g. they could be not densely defined).

1.1.

We first recall some functional analysis terminology and notations. If $\mathcal{H}$ is a Hilbert space then $B(\mathcal{H})$ is the algebra of bounded operators on $\mathcal{H}$, $K(\mathcal{H})$ the ideal of compact operators, $U(\mathcal{H})$ the group of unitary operators, and $1$ or $1_{\mathcal{H}}$ the identity operator. $\mathrm{Sp}(T)$ and $\mathrm{Sp_{ess}}(T)$ are the spectrum and the essential spectrum of an operator $T$.

If $\{\mathcal{A}_{i}\}_{i\in I}$ is a family of linear subspaces of a Banach space $\mathcal{A}$ then

[TABLE]

If $\mathcal{A}_{1},\dots,\mathcal{A}_{n}$ are subsets of a Banach algebra $\mathcal{A}$ then

[TABLE]

By ideal of a $C^{*}$-algebra we mean a closed and self-adjoint ideal. By morphism between $*$-algebras we mean $*$-morphism. We write $\simeq$ for “isomorphic” and $\cong$ for “canonically isomorphic”. If $\mathscr{C}$ is a $C^{*}$-algebra and $\mathcal{P}:\mathscr{C}\to\mathscr{C}$ a morphism such that $\mathcal{P}^{2}=\mathcal{P}$ we say that $\mathcal{P}$ is a projection morphism or morphic projection. Giving a projection morphism is equivalent to giving a direct sum decomposition $\mathscr{C}=\mathscr{A}+\mathscr{I}$ with $\mathscr{A}$ a $C^{*}$-subalgebra and $\mathscr{I}$ an ideal; then $\mathcal{P}$ is the linear projection $\mathscr{C}\to\mathscr{A}$ determined by the direct sum.

We say that a self-adjoint operator $A$ with spectrum $\sigma(A)$ on a Hilbert space $\mathcal{H}$ is affiliated111 This is not the notion introduced by S. L. Woronowicz. to (or with) a $C^{*}$-subalgebra $\mathscr{C}\subset B(\mathcal{H})$ if the next equivalent conditions are satisfied:

[TABLE]

An observable affiliated to an arbitrary $C^{*}$-algebra $\mathscr{C}$ is a morphism $A:C_{0}(\mathbb{R})\to\mathscr{C}$. This notion is discussed in detail in [1, §8.1]. A self-adjoint operator is identified with the observable defined by its $C_{0}\textnormal{-}$functional calculus and we usually denote $\theta(A)$ instead of $A(\theta)$ the value of $A$ at $\theta$ for any observable $A$. We will use the notation

[TABLE]

The zero morphism is an observable affiliated to $\mathscr{C}$ denoted $\infty$; this is natural because $\theta(\infty)=0$ for any $\theta\in C_{0}(\mathbb{R})$. Observables can be described in terms of $\mathscr{C}$-valued self-adjoint resolvents222 The usual terminology is “pseudo-resolvent” but this seems excessive in our setting. the resolvent of $A$ being the map $R_{A}\colon\mathbb{C}\setminus\mathbb{R}\to\mathscr{C}$ with $R_{A}(z)=r_{z}(A)$ where $r_{z}(\lambda)=(\lambda-z)^{-1}$. If $\mathscr{C}$ is realized on a Hilbert space $\mathcal{H}$ then observables may be identified with self-adjoint operators acting in closed subspaces of $\mathcal{H}$ [1, §8.1.2]. For example, the Hamiltonians of N-body systems with hard core interactions are observables affiliated to the $C^{*}$-algebra generated by the usual N-body Hamiltonians [8] but are not self-adjoint operators on $\mathcal{H}$. And $\infty$ is the only operator with domain $\{0\}$.

If $\mathscr{C},\mathscr{D}$ are $C^{*}$-algebras, $\mathcal{P}:\mathscr{C}\to\mathscr{D}$ a morphism, and $A$ an observable affiliated to $\mathscr{C}$, then $\mathcal{P}(A)\doteq\mathcal{P}\circ A$ is an observable affiliated to $\mathscr{D}$. If $\mathscr{C},\mathscr{D}$ are realised on Hilbert spaces $\mathcal{H},\mathcal{K}$ and $A$ is the observable associated to a self-adjoint operator on $\mathcal{H}$ then quite often the observable $\mathcal{P}(A)$ is not associated to a (densely defined) self-adjoint operator on $\mathcal{K}$. Since objects like $\mathcal{P}(A)$ appear naturally in our arguments, we are forced to work with the more general notion of observable instead of self-adjoint operator.

An observable $A$ affiliated to $\mathscr{C}$ is strictly affiliated to $\mathscr{C}$ if the linear span of the operators $\theta(A)T$ with $\theta\in C_{0}(\mathbb{R})$ and $T\in\mathscr{C}$ is dense in $\mathscr{C}$. Then $\mathcal{P}(A)$ is a (densely defined) self-adjoint operator in any non-degenerate representations $\mathcal{P}$ of $\mathscr{C}$ [21, Pr. A.7].

In the case of observables the strong limits are interpreted as follows: if $\{A_{n}\}_{n\in\mathbb{N}}$ is a sequence of observables affiliated to $B(\mathcal{H})$ and $\operatorname*{s-lim}_{n}A_{n}(\theta)\doteq A(\theta)$ exists for any $\theta\in C_{0}(\mathbb{R})$ then $A$ is an observable affiliated to $B(\mathcal{H})$ written $A=\operatorname*{s-lim}_{n}A_{n}$. Each $A_{n}$ could be a densely defined self-adjoint operator without $A$ being so (we may have $A=\infty$).

The $C^{*}$-algebra $C^{*}(\mathcal{A})$ generated by a set $\mathcal{A}$ of observables affiliated to a $C^{*}$-algebra $\mathscr{C}$ is the smallest $C^{*}$-subalgebra of $\mathscr{C}$ to which are affiliated all the $A\in\mathcal{A}$. If $\mathcal{A}=\{A_{i}\mid i\in I\}$ we write $C^{*}(\mathcal{A})=C^{*}(A_{i}\mid i\in I)$. Clearly $C^{*}(A)\equiv C^{*}(\{A\})=\{\theta(A)\mid\theta\in C_{0}(\mathbb{R})\}$. In terms of the resolvents $R_{A}$, for any $z_{0}$ not real $C^{*}(\mathcal{A})$ is the $C^{*}$-subalgebra generated by the operators $R_{A}(z_{0})$ with $A\in\mathcal{A}$. If $A$ is an observable affiliated to $\mathscr{C}$ then $C^{*}(A)$ is the closed linear span of the operators $R_{A}(\mathrm{i}\lambda)$ with $\lambda\in\mathbb{R}\setminus\{0\}$.

If $A_{1},\dots,A_{n}$ are observables with $C^{*}(A_{i})\cdot C^{*}(A_{j})=C^{*}(A_{j})\cdot C^{*}(A_{i})$ and $\mathcal{A}_{I}\doteq C^{*}(A_{i_{1}})\cdot C^{*}(A_{i_{2}})\cdot\ldots\cdot C^{*}(A_{i_{m}})$ for $I=\{i_{1},\dots,i_{m}\}$ not empty subset of $\{1,\dots,n\}$, then $\mathcal{A}_{I}$ is a $C^{*}$-algebra and $C^{*}(A_{1},\dots,A_{n})$ is the closure of $\sum_{I}\mathcal{A}_{I}$.

$C^{*}$-algebras graded by semilattices have been introduced in [6] as a tool in the spectral theory of N-body Hamiltonians and further studied in [1, 7, 19, 20, 21]. An exhaustive study of these algebras may be found in Athina Mageira’s thesis [43, 44]. In this paper we consider only semilattices consisting of finite dimensional subspaces of real vector spaces.

If $\Xi$ is an arbitrary real vector space the Grassmannian $\mathbb{G}(\Xi)$ is the set of its finite dimensional subspaces, the projective space $\mathbb{P}(\Xi)$ the set of its one dimensional subspaces, and $\mathbb{H}(\Xi)$ the set of its hyperplanes. $\mathbb{G}(\Xi)$ is equipped with the order relation given by inclusion: $E\leq F$ means $E\subset F$. Then $\mathbb{G}(\Xi)$ is a lattice: for any pair of its elements $E,F$ their upper bound $E\vee F=E+F$ and lower bound $E\wedge F=E\cap F$ exist in $\mathbb{G}(\Xi)$. We say that $\mathcal{S}\subset\mathbb{G}(\Xi)$ is a subsemilattice if $E,F\in\mathcal{S}\Rightarrow E+F\in\mathcal{S}$. If $X\subset\Xi$ is a vector subspace then $\mathbb{G}(X)$ and $\mathbb{G}_{\scriptscriptstyle\supset X}(\Xi)\doteq\{E\in\mathbb{G}(\Xi)\mid E\supset X\}$ are subsemilattices and also a sublattices of $\mathbb{G}(\Xi)$.

Let $\mathcal{S}\subset\mathbb{G}(\Xi)$ a subsemilattice. A $C^{*}$-algebra $\mathscr{C}$ is $\mathcal{S}$-graded, or graded by $\mathcal{S}$, if a family $\{\mathscr{C}(E)\}_{E\in\mathcal{S}}$ of $C^{*}$-subalgebras of $\mathscr{C}$ is given such that the linear sum $\mathring{\mathscr{C}}\doteq\sum_{E\in\mathcal{S}}\mathscr{C}(E)$ is direct and dense in $\mathscr{C}$ and

[TABLE]

The algebras $\mathscr{C}(E)$ are called components of $\mathscr{C}$. We write $\mathscr{C}=\sum_{E\in\mathcal{S}}^{\mathrm{c}}\mathscr{C}(E)$ meaning that $\mathscr{C}$ is the closure of the linear sum $\mathring{\mathscr{C}}$. The sum $\sum_{E\in\mathcal{S}}\mathscr{C}(E)$ being direct, for any $E\in\mathcal{S}$ each $T\in\mathring{\mathscr{C}}$ has a unique component $T(E)\in\mathscr{C}(E)$ such that $T=\sum_{E}T(E)$. If $\mathcal{T}$ is a subsemilattice with $\mathcal{S}\subset\mathcal{T}$ then an $\mathcal{S}$-graded $C^{*}$-algebra is naturally $\mathcal{T}$-graded.

We sometimes use similarly defined inf-graded $C^{*}$-algebras. A subset $\mathcal{S}\subset\mathbb{G}(\Xi)$ is called stable under intersections, or $\cap$-stable, if $E,F\in\mathcal{S}\Rightarrow E\cap F\in\mathcal{S}$. Then a $C^{*}$-algebra $\mathscr{C}$ is inf-graded by $\mathcal{S}$ if a family $\{\mathscr{C}(E)\}_{E\in\mathcal{S}}$ of $C^{*}$-subalgebras of $\mathscr{C}$ is given such that the sum $\mathring{\mathscr{C}}\doteq\sum_{E\in\mathcal{S}}\mathscr{C}(E)$ is direct and dense in $\mathscr{C}$ and $\mathscr{C}(E)\mathscr{C}(F)\subset\mathscr{C}(E\cap F)\ \forall\ E,F\in\mathcal{S}$.

1.2. Grassmann C*-algebra

The Grassmann $C^{*}$-algebra $\mathcal{G}_{\scriptscriptstyle{X}}$ of a real finite dimensional vector space $X$ is a simple example of inf-graded $C^{*}$-algebra which may be seen as an abelian version of the field algebra. Indeed, besides the inf-grading by $\mathbb{G}(\Xi)$, it is also the $C^{*}$-algebra generated by the linear forms on $X$ which in this context play the rôle of field operators. On the other hand, Grassmann $C^{*}$-algebras also play an important rôle in the description of the structure of the field algebra and in a generalized N-body problem.

We denote $C_{\mathrm{b}}(X)$ the $C^{*}$-algebra of bounded continuous complex functions on $X$ and $C_{\mathrm{b}}^{\mathrm{u}}(X)$, $C_{0}(X)$ and $C_{\mathrm{c}}(X)$ the subalgebras of uniformly continuous functions, functions which tend to zero at infinity, and functions with compact support.

If $Y\in\mathbb{G}(X)$ then we embed $C_{0}(X/Y)\subset C_{\mathrm{b}}^{\mathrm{u}}(X)$ via $\varphi\mapsto\varphi\circ\pi_{Y}$ ($\pi_{Y}$ is the natural surjection of $X$ onto the the quotient space $X/Y$). In other terms, $C_{0}(X/Y)$ is the set of continuous functions $\varphi:X\to\mathbb{C}$ with $\varphi(x+y)=\varphi(x)$ for all $x\in X,y\in Y$ and such that $\varphi(x)\to 0$ when the distance from $x$ to $Y$ (for some norm on $X$) tends to infinity.

The family of $C^{*}$-subalgebras $C_{0}(X/Y)$ with $Y\in\mathbb{G}(X)$ is linearly independent and

[TABLE]

Then $\mathring{\mathcal{G}}_{\scriptscriptstyle{X}}\doteq{\textstyle\sum_{Y\in\mathbb{G}(X)}}C_{0}(X/Y)$ is a unital $*$-subalgebra of $C_{\mathrm{b}}^{\mathrm{u}}(X)$ and

[TABLE]

is a $C^{*}$-subalgebra of $C_{\mathrm{b}}^{\mathrm{u}}(X)$, called (abelian) Grassmann $C^{*}$-algebra of $X$. Clearly $\mathcal{G}_{\scriptscriptstyle{X}}$ is a $C^{*}$-algebra inf-graded by the lattice $\mathbb{G}(X)$ with $C_{0}(X/Y)$ as components, in particular the notation $\mathcal{G}_{\scriptscriptstyle{X}}(Y)=C_{0}(X/Y)$, which we sometimes use, is natural.

There is an alternative description of $\mathcal{G}_{\scriptscriptstyle{X}}$, similar to (1.13), as the $C^{*}$-algebra generated by a simple and natural set of observables affiliated to $C_{\mathrm{b}}^{\mathrm{u}}(X)$. Note first that any uniformly continuous function $u:X\to\mathbb{R}$ may be identified with an observable affiliated to the $C^{*}$-algebra $C_{\mathrm{b}}^{\mathrm{u}}(X)$, namely the morphism $C_{0}(\mathbb{R})\ni\theta\mapsto\theta\circ u\in C_{\mathrm{b}}^{\mathrm{u}}(X)$; in the natural representation of $C_{\mathrm{b}}^{\mathrm{u}}(X)$ on $L^{2}(X)$ this observable is the operator of multiplication by $u$. In particular, any linear form $\varphi:X\to\mathbb{R}$ is an observable affiliated to $C_{\mathrm{b}}^{\mathrm{u}}(X)$, hence the dual vector space $X^{*}$ is a space of such observables.

If $Y\in\mathbb{G}(X)$ let $Y^{\perp}=\{\varphi\in X^{*}\mid\varphi(x)=0\}\in\mathbb{G}(X^{*})$.

Proposition 1.1**.**

We have $\mathcal{G}_{\scriptscriptstyle{X}}=C^{*}(\varphi\mid\varphi\in X^{*})$. If $Y\in\mathbb{G}(X)$ and $\varphi_{1},\dots,\varphi_{n}\in X^{*}$ then $Y=\ker\varphi_{1}\cap\dots\cap\ker\varphi_{n}$ if and only if $\varphi_{1},\dots,\varphi_{n}$ generate $Y^{\perp}$ and then

[TABLE]

The proof is an exercise in linear algebra. The $C^{*}$-algebra of the additive group $X^{*}$ is $C_{0}(X)$ and $C_{0}(X)\subset\mathcal{G}_{\scriptscriptstyle{X}}$, so the group $C^{*}$-algebra is a component of $\mathcal{G}_{\scriptscriptstyle{X}}$. With a terminology à la Buchholz-Grundling $\mathcal{G}_{\scriptscriptstyle{X}}$ would be the resolvent group $C^{*}$-algebra of $X^{*}$ [11].

Remark 1.2**.**

The subalgebra $\mathcal{G}_{\scriptscriptstyle{X}}\subset C_{\mathrm{b}}^{\mathrm{u}}(X)$ is stable under translations by elements of $X$ hence the crossed product $\mathcal{G}_{\scriptscriptstyle{X}}\rtimes X$ is a well defined $C^{*}$-algebra (see the comment after Remark 4.12). We will see in §4.3 that $\mathcal{G}_{\scriptscriptstyle{X}}\rtimes X$ is naturally embedded in the field algebra associated to the symplectic space $T^{*}X=X\oplus X^{*}$ and is the $C^{*}$-algebra generated by the maximal class of N-body type Hamiltonians on $X$ (Theorem 4.13 and Corollary 4.14).

Example 1.3**.**

If $X=\mathbb{R}^{2}$ let $C(\theta)$ be the set of functions $(\alpha,\beta)\mapsto u(\alpha\cos\theta+\beta\sin\theta)$ with $u\in C_{0}(\mathbb{R})$. Then $\{C(\theta)\}_{0\leq\theta<\pi}$ is a linearly independent family of $C^{*}$-subalgebras of $C_{\mathrm{b}}^{\mathrm{u}}(X)$ with $C(\theta_{1})\cdot C(\theta_{2})=C_{0}(X)$ if $\theta_{1}\neq\theta_{2}$ and $\mathcal{G}_{\scriptscriptstyle{X}}=\mathbb{C}+\sum_{0\leq\theta<\pi}^{\mathrm{c}}C(\theta)+C_{0}(X)$.

1.3.

A symplectic space is a real vector space $\Xi$ equipped with a symplectic form, i.e. a bilinear anti-symmetric map $\sigma:\Xi^{2}\to\mathbb{R}$ which is non-degenerate in the following sense: $\sigma(\xi,\eta)=0\ \forall\eta\in\Xi\Rightarrow\xi=0$. If $E\subset\Xi$ we set $E^{\sigma}\doteq\{\xi\in\Xi\mid\sigma(\xi,\eta)=0\ \forall\eta\in E\}$; clearly $E\subset F\Rightarrow F^{\sigma}\subset E^{\sigma}$. We set $(E^{\sigma})^{\sigma}=E^{\sigma\sigma}$. If $E,F$ are finite dimensional subspaces then $E^{\sigma\sigma}=E$ and $E\subset F^{\sigma}\Leftrightarrow F\subset E^{\sigma}$. We have $\cap_{i}E_{i}^{\sigma}=(\sum_{i}E_{i})^{\sigma}$ for any family of subspaces $E_{i}$. A subspace $E$ is isotropic if $E\subset E^{\sigma}$​, coisotropic if $E^{\sigma}\subset E$, Lagrangian if $E=E^{\sigma}$, symplectic if $\sigma$ is non-degenerate on it. $E\in\mathbb{G}(\Xi)$ is symplectic if and only if $E\cap E^{\sigma}=0$. If $\mathbb{G}_{\mathrm{s}}(\Xi)$ is the set of finite dimensional symplectic subspaces

[TABLE]

We refer to [23] for the CCR theory associated to $\Xi$. A representation of $\Xi$ on a Hilbert space $\mathcal{H}$ is a map $W:\Xi\to U(\mathcal{H})$ such that

[TABLE]

The second condition is equivalent to the strong continuity of the restriction of $W$ to finite dimensional subspaces hence $\forall\xi\in\Xi$ we may define the field operator $\phi(\xi)\equiv\phi_{\scriptscriptstyle W}(\xi)$ as the self-adjoint operator such that $W(t\xi)=\mathrm{e}^{\mathrm{i}t\phi(\xi)}$ $\forall t\in\mathbb{R}$. We set $R_{\xi}(z)=(\phi(\xi)-z)^{-1}$.

If $E\in\mathbb{G}(\Xi)$ then $M(E)$ is the space of bounded Borel measures on $E$ and $L^{1}(E)$ the subset of absolutely continuous measures (with $L^{1}(0)=\mathbb{C}\delta_{0}$ if $E=0$).

Definition 1.4**.**

The Kastler $C^{*}$-algebra $\mathscr{K}$ of $\Xi$ in the representation $W$ is the norm closure in $B(\mathcal{H})$ of the set of operators $W(\mu)=\int_{E}W(\xi)\mu(\mathrm{d}\xi)$ with $E\in\mathbb{G}(\Xi)$, $\mu\in M(E)$. The field $C^{*}$-algebra $\mathscr{F}$ of $\Xi$ in the representation $W$ is the norm closure of the set of operators $W(\mu)$ with $E\in\mathbb{G}(\Xi)$ and $\mu\in L^{1}(E)$.

Note that the definition of $\mathscr{F}$ given above is neither that of Buchholz-Grundling nor that of Kastler; see Proposition 1.5 and Remark 2.12 for the equivalence of the definitions.

If $\xi\in\Xi$ then $W(\delta_{\xi})=W(\xi)$ where $\delta_{\xi}$ is the Dirac measure at $\xi$, so $W(\xi)\in\mathscr{K}$; clearly

[TABLE]

$\mathscr{F}\subset\mathscr{K}$ is a $C^{*}$-algebra containing the identity operator. Since $\Xi$ is fixed we generally do not include it in notations but if this is necessary we write $\mathscr{K}(\Xi)$ and $\mathscr{F}^{\scriptscriptstyle\Xi}$. The algebras $\mathscr{K},\mathscr{F}$ associated to different $W$ are canonically isomorphic but if the specification of $W$ or $\Xi$ is necessary we use the notations $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{K}}$, $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}$ or $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{K}}(\Xi)$, $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}$. For example, if $\Xi$ is finite dimensional then by the Stone-Von Neumann theorem one has $\mathcal{H}=\mathcal{H}_{0}\otimes\mathcal{H}_{1}$ and $W=W_{0}\otimes 1_{\mathcal{H}_{1}}$ for some irreducible representation $W_{0}$ of $\Xi$ on $\mathcal{H}_{0}$ hence we have $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{K}}={\prescript{{{\scriptscriptstyle{W}}_{\!0}}\!\!\!}{}{\mathscr{K}}}{}\otimes 1_{\mathcal{K}}$ and $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}={\prescript{{{\scriptscriptstyle{W}}_{\!0}}\!\!\!}{}{\mathscr{F}}}{}\otimes 1_{\mathcal{H}_{1}}$.

Here is a description à la Buchholz and Grundling [10] of $\mathscr{F}$ (see also [40, Th. 23]):

Proposition 1.5**.**

For any $z\in\mathbb{C}\setminus\mathbb{R}$ we have

[TABLE]

Definition 1.6**.**

If $E\in\mathbb{G}(\Xi)$ and $\xi_{1},\dots,\xi_{n}$ is a generating set for $E\in\mathbb{G}(\Xi)$ then

[TABLE]

$\mathscr{F}(E)$* is a $C^{*}$-subalgebra of $\mathscr{F}$ and $\mathscr{F}(0)=\mathbb{C}$.*

Clearly $C^{*}(\phi(\xi))\cdot C^{*}(\phi(\eta))=C^{*}(\phi(\eta))\cdot C^{*}(\phi(\xi))$ $\forall\xi,\eta\in\Xi$ so (1.16) is a $C^{*}$-algebra.

We use the terminology of §1.1 concerning graded $C^{*}$-algebras, but the presentation here is rather self-contained. See §2.6 for the proofs of Theorems 1.7 and 1.10.

Theorem 1.7**.**

The set of $C^{*}$-subalgebras $\mathscr{F}(E)$ of $\mathscr{F}$ has the following properties:

[TABLE]

In particular, the $C^{*}$-algebra $\mathscr{F}$ is graded by the semilattice $\mathbb{G}(\Xi)$ with components $\mathscr{F}(E)$.

Remark 1.8**.**

Note that $\mathscr{F}(E)\cap\mathscr{F}(F)=0$ if $E\subsetneq F$.

By (1.18) and (1.19) $\mathring{\mathscr{F}}$ is a dense unital $*$-subalgebra of $\mathscr{F}$ and we may write

[TABLE]

Since the sum in (1.19) is direct, each $T\in\mathring{\mathscr{F}}$ has, for each $E\in\mathbb{G}(\Xi)$, a unique component $T(E)\in\mathscr{F}(E)$ such that $T=\sum_{E}T(E)$.

Remark 1.9**.**

We will see in Theorem 2.8 that the algebras $\mathscr{F}(E)$ have a rather simple structure: they are isomorphic to tensor products $\mathscr{A}\otimes\mathscr{B}$ with $\mathscr{A}$ an abelian $C^{*}$-algebra and $\mathscr{B}$ isomorphic to the algebra of compact operators on a separable Hilbert space. **

For any $\mathcal{S}\subset\mathbb{G}(\Xi)$ we set

[TABLE]

By (1.20), if $\mathcal{S}$ is finite then $\mathscr{F}(\mathcal{S})=\mathring{\mathscr{F}}(\mathcal{S})$. If $\mathcal{S}$ is a subsemilattice of $\mathbb{G}(\Xi)$, meaning $E,F\in\mathcal{S}\Rightarrow E+F\in\mathcal{S}$, then $\mathscr{F}(\mathcal{S})$ is naturally an $\mathcal{S}$-graded $C^{*}$-algebra (Appendix A).

In particular, any subspace $E\subset\Xi$ determines three $C^{*}$-subalgebras of $\mathscr{F}$:

[TABLE]

Clearly $\mathscr{F}_{\scriptscriptstyle E}$ is a unital $C^{*}$-subalgebra and $\mathscr{F}^{\prime}_{\scriptscriptstyle E}$ and $\mathscr{F}_{\scriptscriptstyle\supset E}$ are ideals of $\mathscr{F}$. From (1.16) we get a description of $\mathscr{F}_{\scriptscriptstyle E}$ in terms of field operators:

[TABLE]

Theorem 1.10**.**

The $C^{*}$-algebra $\mathscr{F}_{\scriptscriptstyle E}$ and the ideal $\mathscr{F}^{\prime}_{\scriptscriptstyle E}$ satisfy

[TABLE]

The projection $\mathcal{P}_{\scriptscriptstyle E}\colon\mathscr{F}\to\mathscr{F}_{\scriptscriptstyle E}$ determined by this direct sum decomposition is a morphism and it is the unique continuous linear map $\mathcal{P}_{\scriptscriptstyle E}\colon\mathscr{F}\to\mathscr{F}$ such that

[TABLE]

For any subspaces $E,F$ we have

[TABLE]

With the terminology introduced in §1.1, $\mathcal{P}_{\scriptscriptstyle E}$ is a projection morphism of $\mathscr{F}$ onto $\mathscr{F}_{\scriptscriptstyle E}$.

1.4.

For $\Xi$ finite dimensional and $W$ irreducible $\mathscr{F}(E)$ has a simple explicit description:

Theorem 1.11**.**

If $\Xi$ is finite dimensional and $W$ is irreducible then $\mathscr{F}(E)$ is the set of $T\in B(\mathcal{H})$ such that:

[TABLE]

This is one of our main results (proved in §3.1). The first condition in (1.28) can be written in the following form, that makes sens even if $\dim\Xi=\infty$:

[TABLE]

If $\Xi$ and $W$ are arbitrary then the conditions (1.28), the first one being written as in (1.29), define an interesting $C^{*}$-algebra $\mathscr{E}(E)$ of operators on $\mathcal{H}$ which contains $\mathscr{F}(E)$, namely the $E$-component of the extended field algebra $\mathscr{E}$ which is graded by $\mathbb{G}(\Xi)$. The field $C^{*}$-algebra $\mathscr{F}$ contains only functions of the fields $\phi(\xi)$. If the representation $W$ is not irreducible then many other physically interesting observables are not affiliated to $\mathscr{F}$, e.g. the Hamiltonians involving spin interactions. The $C^{*}$-algebra $\mathscr{E}$ contains $\mathscr{F}$ and fixes this flaw. We summarize below some facts concerning $\mathscr{E}$ which will be studied in Section 5.

Theorem 1.12**.**

$\mathscr{E}(E)$* is a $C^{*}$-subalgebra of $B(\mathcal{H})$ such that $\mathscr{F}(E)\subset\mathscr{E}(E)$. The family of $C^{*}$-algebras $\{\mathscr{E}(E)\}_{E\in\mathbb{G}(\Xi)}$ is linearly independent and $\mathscr{E}(E)\cdot\mathscr{E}(F)\subset\mathscr{E}(E+F)$.*

Definition 1.13**.**

The extended field $C^{*}$-algebra associated to the representation $W$ is

[TABLE]

and is a $\mathbb{G}(\Xi)$-graded $C^{*}$-algebra of bounded operators on $\mathcal{H}$.

We mention three more facts concerning the relation between $\mathscr{F}$ and $\mathscr{E}$:

1. (1)

$\mathscr{E}(0)=\mathrm{Com}(\Xi)$ where $\mathrm{Com}(\Xi)=\mathrm{Com}^{\scriptscriptstyle{W}}(\Xi)$ is the commutant of the representation $W$, i.e. the set of $T\in B(\mathcal{H})$ such that $[T,W(\xi)]=0\,\forall\xi$. 2. (2)

If $W$ is of finite multiplicity then $\mathscr{E}(E)=\mathscr{F}(E)\cdot\mathrm{Com}(\Xi)$ and $\mathscr{E}=\mathscr{F}\cdot\mathrm{Com}(\Xi)$. 3. (3)

Assume $\Xi$ finite dimensional. Then $\mathscr{E}(\Xi)=\mathscr{F}(\Xi)\cdot\mathrm{Com}(\Xi)$ and $W$ is of finite multiplicity if and only if $\mathscr{E}(\Xi)=K(\mathcal{H})$.

1.5.

The $\mathbb{G}(\Xi)$-graded structure of $\mathscr{F}$ described in Theorem 1.7 is the main tool in the study of $\mathscr{F}$ and its applications. Note that it allows one to construct operators in $\mathscr{F}$ in a straightforward way: for each $E\in\mathbb{G}(\Xi)$ choose $T(E)\in\mathscr{F}(E)$ such that $\sum_{E}T(E)\doteq T$ is norm convergent; then $T\in\mathscr{F}$. Self-adjoint operators affiliated to $\mathscr{F}$ may be constructed by a similar procedure, see for example Theorem A.11 for a general abstract result, we give concrete results later: we will see that in N-body type situations this construction is surprisingly efficient. Note also that, by Theorem 1.11, if $\Xi$ is finite dimensional and $W$ is irreducible then the conditions ensuring the appartenance $T(E)\in\mathscr{F}(E)$ are quite explicit.

Theorem 1.10, which is a straightforward consequence of Theorem 1.7, is important in the spectral analysis of the operators in $\mathscr{F}$ or observables affiliated to $\mathscr{F}$, see Theorem 1.16 and Sections 3 and 5 where we prove extensions of the HVZ theorem describing the essential spectrum of N-body Hamiltonians. We will see that if $\Xi$ is infinite dimensional or is finite dimensional but $W$ is of infinite multiplicity then $\mathscr{F}\cap K(\mathcal{H})=0$, hence if $T\in\mathscr{F}$ then its essential spectrum coincides with its spectrum, so we have nothing to say in these situations. But the relation $\mathscr{F}\cap K(\mathcal{H})=0$ has much worse consequences: it implies that many physically interesting Hamiltonians are not affiliated to $\mathscr{F}$ if $\dim\Xi=\infty$. For this reason in this paper we do not treat examples of infinite dimensional symplectic spaces and of operators affiliated to the corresponding field algebra. But let us make a comment on this case to explain what is missing in $\mathscr{F}$.

We consider the Fock space situation and describe a result from [26]. Let $\Xi$ be a complex infinite dimensional Hilbert space and $\mathcal{H}=\Gamma(\Xi)$ the symmetric Fock space associated to it. We keep the notation $\Xi$ for the underlying real vector space of $\Xi$ equipped with the symplectic structure defined by $\sigma(\xi,\eta)=\Im\langle{\xi}|{\eta}\rangle$. Then $\mathscr{F}$ is a $C^{*}$-algebra of operators on $\mathcal{H}$ which does not contain compact operators and the usual quantum field Hamiltonians are not affiliated to it. The problem comes from the fact that $\Gamma(A)\notin\mathscr{F}$ if $A$ is a bounded operator on the one particle Hilbert space $\Xi$. A solution is to extend $\mathscr{F}$ by adding the necessary free kinetic energies. More precisely, if $\mathcal{O}$ is an abelian $C^{*}$-algebra on the Hilbert space $\Xi$ whose strong closure does not contain compact operators then

[TABLE]

is a $C^{*}$-algebra of operators on $\mathcal{H}$ which contains the compacts and whose quotient with respect to the ideal of compact operators is canonically embedded in $\mathcal{O}\otimes\Phi(\mathcal{O})$ which allows one to describe the essential spectrum of the operators affiliated to $\Phi(\mathcal{O})$. The Hamiltonians of the $P(\varphi)_{2}$ models with a spatial cutoff are affiliated to such algebras. The algebra $\mathscr{A}\doteq\Phi(\mathbb{C})$ has a remarkable property: $K(\mathcal{H})\subset\mathscr{A}$ and $\mathscr{A}/K(\mathcal{H})\cong\mathscr{A}$.

1.6.

From now on until the end of the introduction we assume $\Xi$ finite dimensional. Two finite dimensional symplectic spaces are symplectically isomorphic hence their field algebras are isomorphic and their realizations in irreducible representations are isomorphic.

We may describe $\mathscr{F}_{\scriptscriptstyle E}$ and its commutant in $\mathscr{F}$ independently of the graded structure of $\mathscr{F}$.

Theorem 1.14**.**

*For any subspace $E\subset\Xi$ we have

(1) $\mathscr{F}_{\scriptscriptstyle E}=\{T\in\mathscr{F}\mid[T,W(\xi)]=0\ \forall\xi\in E^{\sigma}\}$,

(2) $\mathscr{F}_{\scriptscriptstyle E^{\sigma}}=\{T\in\mathscr{F}\mid[S,T]=0\ \forall S\in\mathscr{F}_{\scriptscriptstyle E}\}$.*

Corollary 1.15**.**

If $X\in\mathbb{G}(\Xi)$ is Lagrangian then $\mathscr{F}_{X}$ is a maximal abelian subalgebra of $\mathscr{F}$, i.e. if $T\in\mathscr{F}$ then $[S,T]=0$ $\forall S\in\mathscr{F}_{\scriptscriptstyle{X}}$ if and only if $T\in\mathscr{F}_{\scriptscriptstyle{X}}$.

The spectrum of an observable $T$ affiliated to a $C^{*}$-algebra $\mathscr{C}$ is the set $\mathrm{Sp}(T)$ of real $\lambda$ such that $\theta(T)\neq 0$ if $\theta\in C_{0}(\mathbb{R})$ and $\theta(\lambda)\neq 0$. Note that $\infty$ is the only observable with empty spectrum. If $\mathcal{H}$ is a Hilbert space and $\mathscr{C}\subset B(\mathcal{H})$, the essential spectrum of $T$ is the set $\mathrm{Sp_{ess}}(T)$ of $\lambda$ such that $\theta(T)\notin K(\mathcal{H})$ if $\theta(\lambda)\neq 0$. We have

[TABLE]

Indeed, if $E\subset F$ then $\mathcal{P}_{\scriptscriptstyle E}=\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle F}$ and $\mathcal{P}_{\scriptscriptstyle E}$ is a morphism. The next theorem and its corollary follow from Theorem 1.10 via Atkinson’s theorem, cf. Theorems A.5, A.8, 3.4.

Theorem 1.16**.**

If $W$ of finite multiplicity then

[TABLE]

Moreover, for any $H\in\mathbb{H}(\Xi)$ and any nonzero $\xi\in H^{\sigma}$ we have

[TABLE]

The definition of $\mathcal{P}_{\scriptscriptstyle H}T$ and of the limit in (1.34) in the case of observables is given in §1.1. The relation (1.33) can be improved if $T$ belongs to a graded subalgebra of $\mathscr{F}$:

Corollary 1.17**.**

Let $\mathcal{S}\subset\mathbb{G}(\Xi)$ stable under sums with $\Xi\in\mathcal{S}$ and $\mathcal{S}_{\max}$ the set of maximal elements of $\mathcal{S}\setminus\{\Xi\}$. Then

[TABLE]

Indeed, we have $\supset$ in (1.35) by (1.32). On the other hand, if $H$ is a hyperplane let $E\in\mathcal{S}$ of maximal dimension with $E\subset H$. Then $E$ is the greatest element of $\mathcal{S}$ included in $H$ hence $\mathcal{P}_{\scriptscriptstyle E}T=\mathcal{P}_{\scriptscriptstyle H}T$ so $\mathrm{Sp}(\mathcal{P}_{E}T)=\mathrm{Sp}(\mathcal{P}_{H}T)$ for all $T\in\mathscr{F}(\mathcal{S})$.

Remark 1.18**.**

The main points of the proof of Corollary 1.17 are the relation $\mathscr{F}(\Xi)=\mathscr{F}(\mathcal{S})\cap K(\mathcal{H})$ and the fact that the Theorem 1.10 gives us an explicit description of the quotient $\mathscr{F}(\mathcal{S})/\mathscr{F}(\Xi)$ as a $C^{*}$-subalgebra of $\bigoplus_{E\in\mathcal{S}_{\max}}\mathscr{F}_{\scriptscriptstyle E}$ and of the canonical morphism $\mathcal{P}:\mathscr{F}(\mathcal{S})\to\mathscr{F}(\mathcal{S})/\mathscr{F}(\Xi)$. From this one gets (1.35) which is a general version of the HVZ theorem, and also the Mourre estimate with respect to certain conjugate operators by an extension of [1, Th. 8.4.3]. Note that $\mathscr{F}(\mathcal{S})/\mathscr{F}(\Xi)$ is realised on the Hilbert space $\mathcal{H}_{\mathcal{S}}\doteq\bigoplus_{E\in\mathcal{S}_{\max}}\!\mathcal{H}$ and there are many self-adjoint operators $T$ affiliated to $\mathscr{F}(\mathcal{S})$ such that $\mathcal{P}(T)$ is not a densely defined operator on $\mathcal{H}_{\mathcal{S}}$.

1.7.

We now turn to the question of self-adjoint operators are affiliated with the field algebra $\mathscr{F}$. We consider here only the phase space $\Xi=T^{*}\mathbb{R}^{n}$, for a general framework and proofs we refer to Section 4. We will see that $\mathscr{F}$ is rather small and quite simple and natural operators are not affiliated to it, but for example the class of Hamiltonians with an N-body type structure affiliated to $\mathscr{F}$ is surprisingly large. Note that two finite dimensional symplectic spaces of the same dimension are symplectically isomorphic hence their field algebras are isomorphic, so there is no loss of generality in the choice $\Xi=T^{*}\mathbb{R}^{n}$, but this is not natural for some physical systems, e.g. non-relativistic N-body systems with translation invariant potentials require different Euclidean spaces [1, 22].

Let us consider the Euclidean space $X=\mathbb{R}^{n}$ with scalar product $\langle{x},{y}\rangle=\sum_{i=1}^{n}x_{i}y_{i}$. The dual space $X^{*}$ of $X$ is naturally identified with $X$ but it is convenient to keep the notation $X^{*}$ for it. We think of $X$ as the configuration space of a system whose phase space is $\Xi\equiv T^{*}X=X\oplus X^{*}$ with the symplectic form

[TABLE]

The Schrödinger representation [23, §4.2.1] of $\Xi$ acts in $\mathcal{H}=L^{2}(X)$ by the rule

[TABLE]

The Fourier transform is $(Fu)(k)=(2\pi)^{-n/2}\int_{X}\mathrm{e}^{-\mathrm{i}\langle{x},{k}\rangle}u(x)\mathrm{d}x$. The position $q$ and momentum $p$ ($X$-valued) observables are defined as follows: if $\varphi:X\to\mathbb{C}$ is a Borel function then $\varphi(q)$ is the operator of multiplication by $\varphi$ in $\mathcal{H}$ and $\varphi(p)=F^{*}\varphi(q)F$. Let $q_{j}$ be the operator of multiplication by the j-th variable and $p_{j}=-\mathrm{i}\partial_{j}$ with $\partial_{j}$ the derivative with respect to the j-th variable. Then $\langle{x},{p}\rangle=\sum_{j}x_{j}p_{j}$ and $\langle{q},{k}\rangle=\sum_{j}k_{j}q_{j}$ if $x,k\in X$ and the field operator at the point $\xi=(x,k)\in\Xi$ is $\phi(\xi)\doteq\langle{q},{k}\rangle+\langle{x},{p}\rangle$. These operators are self-adjoint if the definitions are conveniently interpreted and the field algebra $\mathscr{F}$ is the $C^{*}$-algebra generated by them. We have

[TABLE]

where $(\mathrm{e}^{\mathrm{i}\langle{q},{k}\rangle}f)(y)=\mathrm{e}^{\mathrm{i}\langle{y},{k}\rangle}f(y)$ and $(\mathrm{e}^{\mathrm{i}\langle{x},{p}\rangle}f)(y)=f(x+y)$. By Definition 1.4 $\mathcal{H}$ is equipped with two remarkable $C^{*}$-algebras: the Kastler algebra $\mathscr{K}$ and the field algebra $\mathscr{F}$. Since $F^{-1}W(x,k)F=W(k,-x)$ we have

[TABLE]

For $T\in B(\mathcal{H})$ we have $W(\xi)^{*}TW(\xi)=\mathrm{e}^{-\mathrm{i}\langle{x},{p}\rangle}\mathrm{e}^{-\mathrm{i}\langle{q},{k}\rangle}T\mathrm{e}^{\mathrm{i}\langle{q},{k}\rangle}\mathrm{e}^{\mathrm{i}\langle{x},{p}\rangle}$ hence the relation (1.12), which is satisfied by any $T\in\mathscr{K}$, is equivalent to

[TABLE]

For example, if $\varphi\in L^{\infty}(X)$ then

[TABLE]

But the analogous result for the field algebra seems quite remarkable (Proposition 4.7):

Theorem 1.19**.**

$\varphi(q)\in\mathscr{F}\Leftrightarrow\varphi(p)\in\mathscr{F}\Leftrightarrow\varphi\in\mathcal{G}_{{\scriptscriptstyle{X}}}$.

$\mathcal{G}_{\scriptscriptstyle{X}}$ is defined in (1.8). If $X=\mathbb{R}$ this means $\varphi\in C_{0}(\mathbb{R})+\mathbb{C}$, hence if $\varphi\in L^{\infty}(\mathbb{R})$ then $\varphi(q)\in\mathscr{F}$ if and only if $\varphi$ is continuous and has finite and equal limits at $\pm\infty$.

We now give examples of simple operators affiliated or not to the field algebra.

Example 1.20**.**

If $X=\mathbb{R}^{2}$ the operators of multiplication by $x^{2}$, $x^{2}+y^{2}$, or $x-y$ are clearly affiliated to $\mathscr{F}$. But that of multiplication by $x^{2}-y^{2}$ is not affiliated to $\mathscr{K}$ because if $\theta\in C_{\mathrm{c}}(\mathbb{R})$ then $\theta(x^{2}-y^{2})$ is not uniformly continuous.**

Example 1.21**.**

The limit in (1.34) is the same as $r\to+\infty$ or $r\to-\infty$, so an anisotropic behaviour is not allowed by $\mathscr{F}$. It follows that certain physically relevant Hamiltonians are not affiliated to $\mathscr{F}$, for example, the one dimensional anisotropic Schrödinger operators. More precisely, let $X=\mathbb{R}$ and $H=p^{2}+v(q)$ with $v$ real locally integrable such that

[TABLE]

Then $(H+\mathrm{i})^{-1}\in\mathscr{F}$ if and only if $a_{+}=a_{-}$ because**

[TABLE]

Similarly, the anisotropic algebras $C(\overline{\mathbb{R}})\cdot C_{0}(\mathbb{R}^{*})$ and $C(\overline{\mathbb{R}})\cdot C(\overline{\mathbb{R}^{*}})$ from [31], although natural algebras of quantum Hamiltonians, are not included in $\mathscr{F}$. The N-body Hamiltonians studied in [34] are not affiliated to the field algebra associated to the natural $X$.**

Example 1.22**.**

This is similar to Example 1.21 but we perturb $p^{2}$ by a second order operator. Let $v\geq 1$ a bounded Borel function on $\mathbb{R}$ satisfying (1.42). Then $H=pv(q)p$, with the Sobolev space $\mathcal{H}^{1}(\mathbb{R})$ as form domain, is affiliated to $\mathscr{F}$ if and only if $a_{+}=a_{-}$.

Example 1.23**.**

For $X=\mathbb{R}$ the Stark Hamiltonian $H=p^{2}+q=\mathrm{e}^{\mathrm{i}p^{3}/3}q\mathrm{e}^{-\mathrm{i}p^{3}/3}$ is not affiliated to $\mathscr{F}$: indeed, if $T=(H+\mathrm{i})^{-1}$ then the second condition in (1.40) is satisfied but not the first one, cf. [27, §4.2.7]. Also $p^{2}-q^{2}$ is not affiliated to $\mathscr{F}$ by [10, Prop. 6.3].**

Example 1.24**.**

If $X=\mathbb{R}^{3}$ and $b$ is a constant magnetic field with magnetic vector potential $a(x)=\frac{1}{2}b\times x$ then the magnetic Hamiltonian $H_{0}=(p-a)^{2}$ is affiliated to $\mathscr{F}$. This is an easy consequence of Theorem 1.11 because $H_{0}=\sum_{j=1}^{3}\phi(\eta_{j})^{2}$ with $\eta_{j}\in T^{*}X$ (Remark 4.4). Theorem A.9 allows one to treat $H=(p-a)^{2}+v$ with rather general $a$ and $v$.

Example 1.25**.**

An example of a different nature is the generator of the dilation group: $\omega=pq+qp=2pq+\mathrm{i}$ is not affiliated to $\mathscr{F}$ (again $X=\mathbb{R}$). Indeed, in Proposition 3.11 we show that $\operatorname*{w-lim}_{|r|\to\infty}W(r\xi)^{*}(\omega+\mathrm{i})^{-1}W(r\xi)=0$ for all $\xi\in\Xi$, $\xi\neq 0$ and $(\omega+\mathrm{i})^{-1}$ is not compact, hence $(\omega+\mathrm{i})^{-1}\notin\mathscr{F}$ by Theorem 1.16. Note that $p^{2}-q^{2}=p^{\prime}q^{\prime}+q^{\prime}p^{\prime}$ where $p^{\prime}=(p-q)/\sqrt{2},q^{\prime}=(p+q)/\sqrt{2}$ satisfy $[p^{\prime},q^{\prime}]=-\mathrm{i}$ **

1.8.

We now consider N-body Hamiltonians in the framework of §1.7. Our purpose is to show that the class of such Hamiltonians affiliated to $\mathscr{F}$ is very large.

We say that a function $h:X\to\mathbb{R}$ is divergent if $\lim_{k\to\infty}h(k)=+\infty$. Below we fix the kinetic energy function $h$ but we allow kinetic energy operators of the form $h(p-k)$ with arbitrary $k\in X$ because the origin of the momentum space should not play a rôle.

Proposition 1.26**.**

Let $h:X\to\mathbb{R}$ be continuous and divergent and $\mathcal{T}_{0}\subset\mathbb{G}(X)$. Denote $\mathcal{T}$ the set of intersections of subspaces from $\mathcal{T}_{0}$ and $\mathcal{T}^{\sigma}=\{Y^{\sigma}\mid Y\in\mathcal{T}\}$, which is a subsemilattice of $\mathbb{G}(\Xi)$. Then $\mathscr{F}(\mathcal{T}^{\sigma})$ is the $C^{*}$-algebra generated by the self-adjoint operators $H=h(p-k)+v(q)$ with $k\in X$ and real $v\in\sum_{Y\in\mathcal{T}_{0}}C_{0}(X/Y)$.

This is a consequence of Theorem 4.13. The proposition says that the algebras $\mathscr{F}(\mathcal{S})$ with $\mathcal{S}$ a subsemilattice of subspaces of $\Xi$ such that $\min\mathcal{S}=X$ are generated by Hamiltonians having an N-body type structure. A computation page 4.15 clarifies this assertion.

Let $\mathcal{S}\subset\mathbb{G}(\Xi)$ be an arbitrary subsemilattice with $\min\mathcal{S}=X$ and $h:X\to\mathbb{R}$ continuous and divergent. Then $h(p)$ is a kinetic energy operator affiliated to $\mathscr{F}(X)=C_{0}(X^{*})$ and our goal is to build self-adjoint operators $H=h(p)+V$ affiliated to $\mathscr{F}(\mathcal{S})$. The case of bounded operators $V$ is easy but not without interest if $\mathcal{S}$ is infinite because we do not assume that $V$ is a sum of components. By Proposition A.10 and Corollary 1.17 we have:

Proposition 1.27**.**

If $V\in\mathscr{F}(\mathcal{S})$ is symmetric then $H=h(p)+V$ is affiliated to $\mathscr{F}(\mathcal{S})$ and $\mathcal{P}_{\scriptscriptstyle E}H=h(p)+\mathcal{P}_{\scriptscriptstyle E}V$ for all $E\in\mathcal{S}$. If $\Xi\in\mathcal{S}$ then

[TABLE]

Remark 1.28**.**

The maximal choice $\mathcal{S}=\mathbb{G}_{\scriptscriptstyle\supset X}$ is particularly interesting, well beyond the N-body problem. We have $\mathcal{S}=\mathcal{T}^{\sigma}$ with $\mathcal{T}\subset\mathbb{G}(X)$ stable under intersections and the N-body case corresponds to finite $\mathcal{T}$: then $H=h(p)+\sum_{Y\in\mathcal{T}}v_{\scriptscriptstyle{Y}}$ with $v_{\scriptscriptstyle{Y}}\in C_{0}(X/Y)$ in the simplest case, cf. (4.27). But if $\mathcal{T}=\mathbb{G}(X)$ then $H=h(p)+V$ and $V$ is not just a sum of $v_{\scriptscriptstyle{Y}}$. In fact, extending from finite $\mathcal{T}$ to $\mathbb{G}(X)$ is like going from trigonometric polynomials to uniformly (Bohr) almost periodic functions. Instead of periodic functions, i.e. functions invariant under translations by elements of subgroups $a\mathbb{Z}$, we use functions invariant under translations by elements of vector subgroups (subspaces) of $X$. Thus the self-adjoint operators affiliated to $\mathscr{F}_{\scriptscriptstyle\supset X}$ could be called “almost N-body Hamiltonians”, meaning that they can be approximated, in some sense, with elementary N-body type Hamiltonians.

In order to treat unbounded $V$ we consider here only functions $h$ such that the form domain of $h(p)$ is a Sobolev space (Theorem 4.16 is a more general but less explicit statement). $\mathcal{H}^{s}\equiv\mathcal{H}^{s}(X)$ is the Sobolev space of order $s\in\mathbb{R}$ defined by the norm $\|u\|_{s}=\|\langle{p}\rangle^{s}u\|$ where $\langle{x}\rangle=(1+|x|^{2})^{1/2}$. We have $\mathcal{H}^{0}=\mathcal{H}$ and if $s\geq 0$ then $\mathcal{H}^{s}\subset\mathcal{H}\subset\mathcal{H}^{-s}=(\mathcal{H}^{s})^{*}$. If $s>0$ is we write $h(x)\sim|x|^{2s}$ if $c^{\prime}|k|^{2s}\leq h(k)\leq c^{\prime\prime}|k|^{2s}$ for some constants $c^{\prime},c^{\prime\prime}$ and all large $k$. If $h$ is bounded from below this is equivalent to $D(|h(p)|^{1/2}=\mathcal{H}^{s}$. Form sums are defined in Appendix §A.4; for the proof of the next theorem see Theorem 4.20.

Theorem 1.29**.**

Let $\mathcal{S}\subset\mathbb{G}(\Xi)$ a subsemilattice with $\min\mathcal{S}=X$ and $h:X\to\mathbb{R}$ continuous with $h(x)\sim|x|^{2s}$ for some $s>0$. Let $V:\mathcal{H}^{s}\to\mathcal{H}^{-s}$ symmetric such that $V\geq-\mu h(p)-\nu$ with $\mu<1,\nu\geq 0$ and $\varphi(p)V\langle{p}\rangle^{-s}\in\mathscr{F}(\mathcal{S})\ \forall\varphi\in C_{\mathrm{c}}(\mathbb{R})$. Then the form sum $H=h(p)+V$ is a self-adjoint operator strictly affiliated to $\mathscr{F}(\mathcal{S})$.

The perturbations $V$ allowed by Theorem 1.29 are much more general than those usually considered in the standard N-body problem even when $\mathcal{S}$ (hence $\mathcal{T}$) is finite and even in the two-body problem (i.e. $\mathcal{T}=\{0,X\}$). We explain this below.

If $s\geq 0$ we denote $\mathbf{B}^{s}(X)\doteq B(\mathcal{H}^{s},\mathcal{H}^{-s})$ and say that $T\in\mathbf{B}^{s}(X)$ is small at infinity if it verifies the following equivalent conditions:

[TABLE]

Let $\mathbf{B}_{0}^{s}(X)$ be the closed subset of $\mathbf{B}^{s}(X)$ consisting of small at infinity operators. For example, $\varphi(q)\in\mathbf{B}^{0}(X)$ if and only if $\varphi\in L^{\infty}(X)$ and $\varphi(q)\in\mathbf{B}_{0}^{0}(X)$ means that we also have $\lim_{a\to\infty}\int_{|x-a|}|\varphi(x)|\mathrm{d}x=0$ (which explains the conditions (1.42)).

If $Y\in\mathbb{G}(X)$ and $Y^{\prime}$ is a complementary subspace of $Y$ in $X$ then $X=Y\oplus Y^{\prime}$ and

[TABLE]

Let $F_{\scriptscriptstyle{Y}}$ be the Fourier transform in $L^{2}(Y)$ and keep the notation $F_{\scriptscriptstyle{Y}}$ for $F_{\scriptscriptstyle{Y}}\otimes 1$ acting in $L^{2}(X)=L^{2}(Y)\otimes L^{2}(Y^{\prime})$. Let $q_{\scriptscriptstyle{Y}},p_{\scriptscriptstyle{Y}}$ be the position and momentum observables associated to $Y$, so $\varphi(p_{\scriptscriptstyle{Y}})=F_{\scriptscriptstyle{Y}}^{-1}\varphi(q_{\scriptscriptstyle{Y}})F_{\scriptscriptstyle{Y}}$ if $\varphi:Y\to\mathbb{C}$ and $C_{0}(Y^{*})=F_{\scriptscriptstyle{Y}}^{-1}C_{0}(Y)F_{\scriptscriptstyle{Y}}$ (cf. §1.7 and §4.1). If $\Phi:Y\to\mathbf{B}(Y^{\prime})$ is continuous and we denote $\Phi(q_{\scriptscriptstyle{Y}})$ the operator of multiplication by $\Phi$ in $L^{2}(X)=L^{2}(Y;L^{2}(Y^{\prime}))$, then $\Phi(p_{\scriptscriptstyle{Y}})\doteq F_{\scriptscriptstyle{Y}}^{-1}\Phi(q_{\scriptscriptstyle{Y}})F_{\scriptscriptstyle{Y}}$.

This definition extends to functions on $Y$ with values operators on Sobolev spaces on $Y^{\prime}$. For example, if $V_{\scriptscriptstyle{Y}}:Y\to\mathbf{B}^{s}(Y^{\prime})$ is continuos and $V_{\scriptscriptstyle{Y}}(y):\mathcal{H}^{s}(Y^{\prime})\to\mathcal{H}^{-s}(Y^{\prime})$ is symmetric and satisfies $\pm V_{\scriptscriptstyle{Y}}(y)\leq c_{\scriptscriptstyle{Y}}\langle{|y|+|p_{Y^{\prime}}|}\rangle^{2s}$ for some constant $c_{\scriptscriptstyle{Y}}$ and all $y$, then clearly $\pm V_{\scriptscriptstyle{Y}}(p_{\scriptscriptstyle{Y}})\leq C_{\scriptscriptstyle{Y}}\langle{p}\rangle^{2s}$ for some number $C_{\scriptscriptstyle{Y}}$, hence $V_{\scriptscriptstyle{Y}}(p_{\scriptscriptstyle{Y}})\in\mathbf{B}^{s}(X)$.

Let $\mathcal{T}\subset\mathbb{G}(X)$ a finite $\cap$-stable subset with $0,X\in\mathcal{T}$. Denote $\mathcal{S}=\mathcal{T}^{\sigma}$ and $\mathcal{S}_{\max}$ the set of maximal elements of $\mathcal{S}\setminus\{\Xi\}$.

Theorem 1.30**.**

For any $Y\in\mathcal{T}$ let $Y^{\prime}$ a complementary subspace of $Y$ in $X$ and $V_{\scriptscriptstyle{Y}}$ a norm continuos function $Y\to\mathbf{B}_{0}^{s}(Y^{\prime})$ such that $V_{\scriptscriptstyle{Y}}(y):\mathcal{H}^{s}(Y^{\prime})\to\mathcal{H}^{-s}(Y^{\prime})$ is symmetric and satisfies $\pm V_{\scriptscriptstyle{Y}}(y)\leq c_{\scriptscriptstyle{Y}}\langle{|y|+|p_{Y^{\prime}}|}\rangle^{2s}$ for a constant $c_{\scriptscriptstyle{Y}}$ and all $y\in Y$. Let $V_{0}=0$. For $E=Y^{\sigma}\in\mathcal{S}$ assume that the symmetric operator $V(E)=V_{\scriptscriptstyle{Y}}(p_{\scriptscriptstyle{Y}})\in\mathbf{B}^{s}(X)$ satisfies:

[TABLE]

Then the form sums $H=h(p)+\sum_{E}V(E)$ and $H_{\scriptscriptstyle E}=h(p)+\sum_{F\leq E}V(F)$ are bounded from below self-adjoint operators, with form domain $\mathcal{H}^{s}$, strictly affiliated to $\mathscr{F}(\mathcal{S})$, and $\mathcal{P}_{\scriptscriptstyle E}H=H_{\scriptscriptstyle E}\ \forall E\in\mathcal{S}$. Moreover, we have

[TABLE]

We mention that the relations (1.43) and (1.44) are HVZ type theorems for $H$.

Remark 1.31**.**

Let us make a formal comment on the structure of the “potentials” $V_{\scriptscriptstyle{Y}}(p_{\scriptscriptstyle{Y}})$ to clarify the gain in generality compared to potentials $V_{\scriptscriptstyle{Y}}$ in the standard N-body problem. The usual $V_{\scriptscriptstyle{Y}}$ is a function of the internal variable $q_{\scriptscriptstyle Y^{\prime}}$ (of class $L^{u}$ for the natural number $u$) while $V_{\scriptscriptstyle{Y}}(p_{\scriptscriptstyle{Y}})$ is an operator depending on $q_{\scriptscriptstyle Y^{\prime}}$ and on the total momentum $p=(p_{\scriptscriptstyle{Y}},p_{\scriptscriptstyle Y^{\prime}})$.

Example 1.32**.**

The simplest case is $\mathcal{T}=\{0,X\}$, hence $\mathcal{S}=\{X,\Xi\}$ and $\mathcal{F}(\mathcal{S})=C_{0}(X^{*})+\mathbf{K}(X)$. Then $V(X)=0$ hence $V=V(\Xi)$. Since $0^{\sigma}=\Xi$ we have $Y=0$ hence $Y^{\prime}=X$ so $V:\mathcal{H}^{s}\to\mathcal{H}^{-s}$ must be a symmetric small at infinity operator such that $V\geq-\mu h(p)-\nu$ with $\mu<1$ (the operator $V$ may have the same order $2s$ as $h(p)$). Then $H=h(p)+V$ is strictly affiliated to $\mathcal{F}(\mathcal{S})$ and $\mathrm{Sp_{ess}}(H)=h(X)=[\min h,\infty[$.

2. Field algebra

2.1. Kastler algebra

We recall here some basic facts about Kastler’s algebra [40].

2.1.1. Measure spaces

If $X$ is a finite dimensional real vector space we denote $M(X)$ the space of bounded Borel measures on $X$ identified with the Banach space dual of $C_{0}(X)$ and $L^{1}(X)$ the subspace of absolutely continuous measures. If $X=0\equiv\{0\}$ then $M(X)=L^{1}(X)=\mathbb{C}\delta_{0}\equiv\mathbb{C}$ where $\delta_{0}$ is the Dirac measure at zero. If $Y\subset X$ is a subspace then $M(Y)$ is identified with the closed subspace of $M(X)$ consisting of the measures with support in $Y$, so $M(Y)\subset M(X)$ isometrically, and $L^{1}(Y)\subset M(X)$ consists of measures with support in $Y$ and whose restriction to $Y$ is absolutely continuous. Since the elements of $L^{1}(X)$ are thought as measures not functions their norms do not depend of the choice of a Lebesgue measure on $X$. $\mathbb{S}(X)$ is the space of Schwartz test functions with dual the space $\mathbb{S}^{\prime}(X)$ of tempered distributions, hence $L^{1}(X)\subset M(X)\subset\mathbb{S}^{\prime}(X)$ continuously. We set $\langle{u},{v}\rangle=v(u)$ if $u\in\mathbb{S}(X),v\in\mathbb{S}^{\prime}(X)$.

2.1.2. Definition

If $\Xi$ is a symplectic space then $\{M(E)\}_{E\in\mathbb{G}(\Xi)}$ is a filtered family of Banach spaces with $M(E)\subset M(F)$ isometrically if $E\subset F$ so $M(\Xi)=\cup_{E\in\mathbb{G}(\Xi)}M(E)$ is a normed vector space; we denote $\|\cdot\|_{1}$ its norm. If $E,F\in\mathbb{G}(\Xi)$ and $E\neq F$ then

[TABLE]

According to Kastler, the twisted convolution algebra of $\Xi$ is the normed space $M(\Xi)$ equipped with the following unital $*$-algebra structure: if $\mu,\nu\in M(\Xi)$ then $\exists\,E\in\mathbb{G}(\Xi)$ such that $\mu,\nu\in M(E)$ and then the twisted convolution $\mu\oast\nu\in M(E)$ is defined by

[TABLE]

the unit is the Dirac measure at zero $\delta_{0}$, and the adjoint measure $\mu^{*}$ is defined by

[TABLE]

If $\Xi$ is finite dimensional then Kastler [40, Th. 15] proves that the $*$-algebra $M(\Xi)$ has, modulo equivalence, only one faithful irreducible representation and any faithful representation of $M(\Xi)$ is a multiple of this one; in particular, $M(\Xi)$ has a unique $C^{*}$-norm $\|\cdot\|$ and $\|\mu\|\leq\|\mu\|_{1}=\int_{\Xi}|\mu|$. The $C^{*}$-algebra completion $\mathscr{M}(\Xi)$ of $M(\Xi)$ will be called Kastler $C^{*}$-algebra of $\Xi$. Moreover, by [42, Th. 8] we have a continuous embedding

[TABLE]

If $\Xi$ is infinite dimensional we have (1.10) and Kastler’s theorem implies

[TABLE]

hence one may define its Kastler $C^{*}$-algebra as

[TABLE]

From now on, since $\Xi$ is fixed, we simplify the notation and set $\mathscr{M}=\mathscr{M}(\Xi)$. We keep the notations $\mu,\nu$ for the elements of $\mathscr{M}$ and $\mu\oast\nu$ for their product. Then for any $E\in\mathbb{G}(\Xi)$ we have $E\subset F$ for some $F\in\mathbb{G}_{\mathrm{s}}(\Xi)$ hence $M(E)\subset M(F)\subset\mathscr{M}$ and

[TABLE]

is a $C^{*}$-subalgebra of $\mathscr{M}$ and if $E$ is symplectic $\mathscr{M}(E)$ is canonically isomorphic to the Kastler algebra of $E$, so depends only on the restriction of $\sigma$ to $E$. We have (see page 2.11)

[TABLE]

For any $\xi\in\Xi$ let $\delta_{\xi}$ be the Dirac measure at $\xi$ thought as element $\delta_{\xi}\in M(E)\subset\mathscr{M}$ for any $E\in\mathbb{G}(\Xi)$ such that $\xi\in E$. If $\xi,\eta\in\Xi$ we clearly have

[TABLE]

Thus $\delta_{\xi}$ is a unitary element of $\mathscr{M}$ hence $\|\delta_{\xi}\|=1$. Let $\xi_{\sigma}\doteq\sigma(\cdot,\xi)\colon\Xi\to\mathbb{R}$ and $\tau_{\xi}$ the translation by $\xi$ defined by $(\tau_{\xi}f)(\eta)=f(\eta-\xi)$. Clearly, if $E\in\mathbb{G}(\Xi)$ and $\mu\in M(E)$

[TABLE]

If $f:\Xi\to\mathbb{R}$ we write $\lim_{\xi\to 0}f(\xi)=0$ if this holds for the restriction of $f$ to any finite dimensional subspace of $\Xi$. Then by a density and continuity argument we easily get

[TABLE]

2.1.3. Representations

If we embed $\Xi\subset\mathscr{M}(\Xi)$ by using the map $\xi\mapsto\delta_{\xi}$ then the representations of $\Xi$ naturally extend to representations of the Kastler algebra $\mathscr{M}(\Xi)$. Proposition 2.2 is a simple consequence of Stone-Von Neumann’s and Kastler’s theorems.

Remark 2.1**.**

We recall the statement Stone-Von Neumann theorem [23, 25] because we will need several times the notations introduced here: any finite dimensional $\Xi$ has an irreducible representation $W_{0}:\Xi\to U(\mathcal{H}_{0})$ and if $W:\Xi\to U(\mathcal{H})$ is an arbitrary representation then there is a Hilbert space $\mathcal{K}$ and a unitary operator $V:\mathcal{H}\to\mathcal{H}_{0}\otimes\mathcal{K}$ such that $VW(\xi)V^{*}=W_{0}(\xi)\otimes 1_{\mathcal{K}}$ $\forall\ \xi\in\Xi$. The multiplicity of $W$ is the dimension of $\mathcal{K}$ and the commutant of $W$ is the set $\mathrm{Com}^{\scriptscriptstyle{W}}(\Xi)=\{T\in B(\mathcal{H})\mid[T,W(\xi)]=0\,\forall\xi\}$. Clearly $V\mathrm{Com}^{\scriptscriptstyle{W}}(\Xi)V^{*}=1_{\mathcal{H}_{0}}\otimes B(\mathcal{K})$.

Proposition 2.2**.**

If $W:\Xi\to U(\mathcal{H})$ is a representation of the symplectic space $\Xi$ then there is a unique $C^{*}$-algebra representation $W:\mathscr{M}(\Xi)\to B(\mathcal{H})$ such that

[TABLE]

This representation is faithful.

Proof.

First assume $\Xi$ finite dimensional and define $W:M(\Xi)\to B(\mathcal{H})$ by (2.12) with $E=\Xi$. We have $W(\mu\oast\nu)=W(\mu)W(\nu)$, $W(\mu^{*})=W(\mu)^{*}$ and $W(\delta_{\xi})=W(\xi)$ hence $W:M(\Xi)\to B(\mathcal{H})$ is a unital morphism such that $\|W(\mu)\|\leq\|\mu\|_{1}$. To prove that this morphism is injective we use the Stone-Von Neumann theorem with the notations used above. From $VW(\xi)V^{*}=W_{0}(\xi)\otimes 1_{\mathcal{K}}$ we get $VW(\mu)V^{*}=W_{0}(\mu)\otimes 1_{\mathcal{K}}$ hence $\|W(\mu)\|=\|W_{0}(\mu)\|\ \forall\mu\in M(\Xi)$. Hence it suffices to find one representation of $\Xi$ whose associated morphism is injective; this is the case for the regular representation [40, Th. 5].

Since $W:M(\Xi)\to B(\mathcal{H})$ is a faithful representation the relation $\|\mu\|_{0}=\|W(\mu)\|$ defines a $C^{*}$-norm on $M(\Xi)$ hence by Kastler’s theorem we see that $\|W(\mu)\|=\|\mu\|$ and this clearly implies the statement of the theorem if $\Xi$ is finite dimensional.

If $\Xi$ is infinite dimensional and $F\in\mathbb{G}_{\mathrm{s}}(\Xi)$ the restriction $W|F$ is a representation of $F$ so it extends to an isometric representation $W_{F}\colon\mathscr{M}(F)\to B(\mathcal{H})$. If $F,G,H\in\mathbb{G}_{\mathrm{s}}(\Xi)$ and $F+G\subset H$ then $W_{H}|\mathscr{M}(F)=W_{F}$ and $W_{H}|\mathscr{M}(G)=W_{G}$ so $W$ induces a $C^{*}$-algebra representation $\mathscr{M}(\Xi)\to B(\mathcal{H})$ that we also denote $W$ which clearly satisfies (2.12). Since the restriction of $W$ to any $\mathscr{M}(F)$ is isometric and $\mathscr{M}(\Xi)$ is the closure of the union of such $\mathscr{M}(F)$, it follows that the $W\colon\mathscr{M}(\Xi)\to B(\mathcal{H})$ is isometric. ∎

Definition 2.3**.**

The Kastler $C^{*}$-algebra in the representation $W$ is the norm closure $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{K}}(\Xi)$ of the set of operators $W(\mu)$ with $\mu\in M(\Xi)$. The set $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{K}}(\Xi)$ is a unital $C^{*}$-subalgebra of $B(\mathcal{H})$ and $\mu\mapsto W(\mu)$ extends to an isomorphism $W\colon\mathscr{M}(\Xi)\to\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{K}}(\Xi)$.

Remark 2.4**.**

By Proposition 2.2, if $W_{1},W_{2}$ are representations of $\Xi$ then the algebras $\prescript{{{\scriptscriptstyle{W}}_{\!1}}\!\!\!}{}{\mathscr{K}}(\Xi)$ and $\prescript{{{\scriptscriptstyle{W}}_{\!2}}\!\!\!}{}{\mathscr{K}}(\Xi)$ are canonically isomorphic in the following sense: there is a unique morphism $\prescript{{{\scriptscriptstyle{W}}_{1}}\!\!\!}{}{\mathscr{K}}(\Xi)\to\prescript{{{\scriptscriptstyle{W}}_{2}}\!\!\!}{}{\mathscr{K}}(\Xi)$ which sends $W_{1}(\xi)$ into $W_{2}(\xi)$ for all $\xi\in\Xi$. For this reason* we drop the superscript $W$ and write $\mathscr{K}(\Xi)$ or just $\mathscr{K}$ unless $W$ plays a role.*

Remark 2.5**.**

If $\dim\Xi<\infty$ then $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{K}}(\Xi)\cong\prescript{{{\scriptscriptstyle{W}}_{0}}\!\!\!}{}{\mathscr{K}}\otimes 1_{\mathcal{K}}$ with the notations of Remark 2.1.**

2.1.4. C*-norm on $L^{1}$

If $E\in\mathbb{G}(\Xi)$ is not symplectic the $C^{*}$-algebra $\mathscr{M}(E)$ is defined in (2.7) but the $C^{*}$-norm on $M(E)$ is not unique, it depends on the embedding of $E$ in $\Xi$. But the situation is better if we replace $M(E)$ by $L^{1}(E)$: there is a unique $C^{*}$-norm on $L^{1}(E)$ without non-degeneracy condition on $\sigma$. This is interesting since it may be used to construct the field algebra without referring to Kastler algebra (cf. comment after (2.15)).

Theorem 2.6**.**

Let $\Xi$ be a finite dimensional real vector space equipped with a Lebesgue measure and a bilinear anti-symmetric form $\sigma:\Xi\times\Xi\to\mathbb{R}$. Equip $L^{1}(\Xi)$ with the Banach $*$-algebra structure defined by the involution $f^{*}(\xi)=\bar{f}(-\xi)$ and the twisted convolution

[TABLE]

Then there is a unique $C^{*}$-norm on $L^{1}(\Xi)$ (also independent of the chosen measure $\mathrm{d}\eta$).

Proof.

If $\sigma$ is non-degenerate the assertion follows from Stone-Von Neumann theorem and if $\sigma=0$ then this holds for arbitrary locally compact abelian groups, cf. [5, p. 224].

We will need the following theorem of B.A. Barnes [4, Thm. 5.5]. Recall that a Banach $*$-algebra $\mathcal{A}$ is postliminal if its enveloping $C^{*}$-algebra $\tilde{\mathcal{A}}$ is postliminal, i.e. the range of each irreducible representation of $\tilde{\mathcal{A}}$ contains a nonzero compact operator (then it will contain all compact operators) [24, §4.1.10 and Theorem 9.1]. The theorem says that if $\mathcal{A},\mathcal{B}$ are postliminal Banach $*$-algebras and have unique $C^{*}$-norms, then the algebraic tensor product $\mathcal{A}\otimes\mathcal{B}$ has a unique $C^{*}$-norm. Note that the convolution algebra of an amenable group can have several distinct $C^{*}$-norms [4, p. 1].

Let $\Xi^{\sigma}=\{\xi\in\Xi\mid\sigma(\xi,\eta)=0\ \forall\eta\in\Xi\}$ and $\Theta$ a subspace supplementary to $\Xi^{\sigma}$ in $\Xi$, so $\Xi=\Xi^{\sigma}\oplus\Theta$. Note that $\sigma=0$ on $\Xi^{\sigma}$. The restriction of $\sigma$ to $\Theta$ is non-degenerate because if $\xi\in\Theta$ and $\sigma(\xi,\eta)=0$ for all $\eta\in\Theta$ then $\sigma(\xi,\eta)=0$ for all $\eta\in\Xi$ hence $\xi\in\Xi^{\sigma}$ so $\xi=0$. Thus $\Theta$ is a symplectic space when equipped with $\sigma|_{\Theta^{2}}$.

We have $L^{1}(\Xi)=L^{1}(\Xi^{\sigma})\hat{\otimes}L^{1}(\Theta)$ where $\hat{\otimes}$ denote the projective tensor product. It is clear that this equality holds at the Banach $*$-algebra level if we equip $L^{1}(\Xi)$ with the twisted convolution product (2.13), $L^{1}(\Xi^{\sigma})$ with the usual convolution product, and $L^{1}(\Theta)$ with the twisted convolution associated to its symplectic form $\sigma|_{\Theta^{2}}$.

We apply now Barnes’ theorem to the algebraic tensor product $L^{1}(\Xi)=L^{1}(\Xi^{\sigma})\otimes L^{1}(\Theta)$. The $C^{*}$-envelop of $L^{1}(\Xi^{\sigma})$ is the abelian algebra of $C_{0}$ functions on the dual of $\Xi^{\sigma}$, hence is postliminal. Moreover, as we noticed at the beginning of the proof, there is only one $C^{*}$-norm on $L^{1}(\Xi^{\sigma})$. On the other hand, the Stone-Von Neumann theorem applied to the symplectic space $\Theta$ implies that the $C^{*}$-envelop of $L^{1}(\Theta)$ is the $C^{*}$-algebra of compact operators on the Hilbert space of the unique (modulo unitary equivalence) irreducible representation of $\Theta$, hence $L^{1}(\Theta)$ is postliminal and has a unique $C^{*}$-norm. Thus there is a unique $C^{*}$-norm on $L^{1}(\Xi^{\sigma})\otimes L^{1}(\Theta)$ hence on its projective completion $L^{1}(\Xi)$. ∎

2.2. Components of the field algebra

Let $\Xi$ be a symplectic space. If $E\in\mathbb{G}(\Xi)$ then $L^{1}(E)$ is an ideal of $M(E)$ so its closure in $\mathscr{M}(E)$ is a $C^{*}$-subalgebra and an ideal of $\mathscr{M}(E)$ that we denote

[TABLE]

In particular $\mathscr{L}(0)=\mathbb{C}$. By Theorem 2.6 we have

[TABLE]

so we could use Mageira’s reconstruction theorem [43, Th. 2.2] to define the field algebra independently of the Kastler algebra. However, viewing it as a subalgebra of $\mathscr{M}(\Xi)$ could be useful for further developments, e.g. for the treatment of anisotropic operators mentioned in Example 1.21. In this subsection we study the structure of the algebras $\mathscr{L}(E)$.

If $\dim\Xi<\infty$ we have a simple intrinsic description of $\mathscr{L}(E)$:

Theorem 2.7**.**

If $\Xi$ is finite dimensional then $\mathscr{L}(E)$ is the set of $\mu\in\mathscr{M}$ such that

[TABLE]

This is a corollary of Theorem 1.11 which is a much deeper fact because it does not involve a condition like $\mu\in\mathscr{M}$ where $\mathscr{M}$ is a “complicated” algebra.

We call center of $E\in\mathbb{G}(\Xi)$ the space $E_{\mathrm{c}}=E\cap E^{\sigma}=\{\xi\in E\mid\sigma(\xi,\eta)=0\ \forall\eta\in E\}$. Clearly $E/E_{\mathrm{c}}$ is naturally a symplectic space, so the $C^{*}$-algebra $\mathscr{L}(E/E_{\mathrm{c}})$ is well defined. $E$ is symplectic if and only if $E_{\mathrm{c}}=0$, $E$ is isotropic if and only if $E_{\mathrm{c}}=E$, and $E$ is coisotropic if and only if $E_{\mathrm{c}}=E^{\sigma}$.

If $E^{*}$ is the dual of $E$ then $\Xi\ni\xi\mapsto\xi_{\sigma}|_{E}\in E^{*}$ is surjective with kernel $E^{\sigma}$ hence we may identify $E^{*}=\Xi/E^{\sigma}$. Thus if $\Xi$ is finite dimensional $C_{0}(E^{*})\subset C_{\mathrm{b}}^{\mathrm{u}}(\Xi)\subset\mathbb{S}^{\prime}(\Xi)$.

A $C^{*}$-algebra $\mathcal{A}$ is called elementary if it is isomorphic to the $C^{*}$-algebra of compact operators on a separable Hilbert space. The $C^{*}$-tensor products at (3) and (4) below are unambiguously defined, both factors being nuclear algebras.

Theorem 2.8**.**

*Let $E$ be a finite dimensional subspace of $\Xi$.

(1) $E$ is isotropic if and only $\mathscr{L}(E)$ is abelian and then $\mathscr{L}(E){\cong}C_{0}(E^{*})$.

(2) $E$ is symplectic if and only if $\mathscr{L}(E)$ is elementary.

(3) $\mathscr{L}(E)\simeq\mathscr{L}(E_{\mathrm{c}})\otimes\mathscr{L}(E/E_{\mathrm{c}})$.

(4) If $E$ is coisotropic then $\mathscr{L}(E)\simeq\mathscr{L}(E^{\sigma})\otimes\mathscr{L}(E/E^{\sigma})$.

(5) $\mathscr{L}(E)$ is unital if and only if $E=0$ and then $\mathscr{L}(0)=\mathbb{C}$.*

Proof.

$\mathscr{L}(E)$ is abelian if and only $\mu\oast\nu=\nu\oast\mu$ $\forall\mu,\nu\in L^{1}(E)$ which by (2.2) means

[TABLE]

Integrating over $\xi$ we then get

[TABLE]

hence $\mathrm{e}^{-\frac{\mathrm{i}}{2}\sigma(\xi,\eta)}-\mathrm{e}^{\frac{\mathrm{i}}{2}\sigma(\xi,\eta)}=0\ \forall\xi,\eta\in E$ which clearly is equivalent to $\sigma|_{E^{2}}=0$.

We now describe the canonical isomorphism $\mathscr{L}(E)\cong C_{0}(E^{*})$ for isotropic $E$. Then for $\mu,\nu\in L^{1}(E)$ the twisted convolution product $\mu\oast\nu$ coincides with the ordinary convolution product $\mu\star\nu$. Thus $L^{1}(E)$ is the usual Banach $*$-algebra associated to the convolution product on $E$ whose enveloping $C^{*}$-algebra is identified with $C_{0}(E^{*})$ via a Fourier transformation. We still need to show that the norm induced by $\mathscr{M}(\Xi)$ on $L^{1}(E)$ coincides with the norm induced by the enveloping $C^{*}$-algebra. But this follows from the fact that for an arbitrary locally compact abelian group $E$ there is only one $C^{*}$-norm on the convolution algebra $L^{1}(E)$ [5, p. 224]. This proves (1).

If $E$ is symplectic then by definition $\mathscr{L}(E)$ is the completion of the twisted convolution algebra $L^{1}(E)$ in $\mathscr{M}(E)$. If $W$ is an irreducible representation of $E$ then from Proposition 2.2 and the Stone-Von Neumann theorem it follows that $\mathscr{L}(E)$ is isomorphic to the $C^{*}$-algebra generated by the operators $W(\mu)=\int_{E}W(\xi)\mu(\mathrm{d}\xi)$ with $\mu\in L^{1}(E)$, which is the $C^{*}$-algebra of compact operators on the Hilbert space of the representation (obvious in the Schrödinger representation). Reciprocally, if $E$ is not symplectic then $E_{\mathrm{c}}\neq 0$ hence $\mathscr{L}(E)$ is not elementary by (3).

Assertion (3) follows from Theorem 2.6 and its proof. If $F$ is a subspace supplementary to $E_{\mathrm{c}}$ in $E$, so $E=E_{\mathrm{c}}\oplus F$, then $L^{1}(E)=L^{1}(E_{\mathrm{c}})\hat{\otimes}L^{1}(F)$ projective tensor product of the usual convolution algebra $L^{1}(E_{\mathrm{c}})$ with the twisted convolution algebra $L^{1}(F)$ defined by the symplectic structure of $F$. By the uniqueness of the $C^{*}$-norms of these algebras and [4, Thm. 5.5] we get $\mathscr{L}(E)=\mathscr{L}(E_{\mathrm{c}})\otimes\mathscr{L}(F)$ which is the uniquely defined $C^{*}$-tensor product. This proves (3). Finally, (4) is a particular case of (3) and (5) is true because by the preceding description the algebras $\mathscr{L}(E)$ with $E\neq 0$ cannot have unit. ∎

2.3. Field algebra

The set of $C^{*}$-subalgebras $\mathscr{L}(E)$ of $\mathscr{M}$ has some remarkable properties summarised in the next theorem. If $\mathcal{A}_{1},\dots,\mathcal{A}_{n}$ are subsets of $\mathscr{M}$ then $\mathcal{A}_{1}\oast\dots\oast\mathcal{A}_{n}$ is the linear span of the products $\mu_{1}\oast\dots\oast\mu_{n}$ with $\mu_{i}\in\mathcal{A}_{i}$ and $\mathcal{A}_{1}\dot{\oast}\dots\dot{\oast}\mathcal{A}_{n}$ its closure.

Theorem 2.9**.**

(1)* The family of linear subspaces $\mathscr{L}(E)$ of $\mathscr{M}$ is linearly independent.

(2) If $E,F\in\mathbb{G}(\Xi)$ and $\mu\in\mathscr{L}(E),\nu\in\mathscr{L}(F)$ then $\mu\oast\nu\in\mathscr{L}(E+F)$.

(3) $\mathscr{L}(E)\dot{\oast}\mathscr{L}(F)=\mathscr{L}(E+F)$.

(4) If $\mathcal{S}\subset\mathbb{G}(\Xi)$ is finite then $\sum_{E\in\mathcal{S}}\mathscr{L}(E)$ is norm closed.*

From (1) and (2) it follows that the linear subspace generated by the $\mathscr{L}(E)$, denoted

[TABLE]

is a unital $*$-subalgebra of $\mathscr{M}$ hence its closure is a unital $C^{*}$-subalgebra. The sum in (2.17) is direct hence if $\mu\in\mathring{\mathscr{L}}$ its component in $\mathscr{L}(E)$ is uniquely determined; we denote it $\mu(E)$, so that $\mu=\sum_{E}\mu(E)$.

Definition 2.10**.**

The field $C^{*}$-algebra $\mathscr{L}\equiv\mathscr{L}^{{\scriptscriptstyle\Xi}}$ of $\Xi$ is the closure of $\mathring{\mathscr{L}}$:

[TABLE]

Parts (1) and (2) of Theorem 2.9 and the preceding definition say that $\mathscr{L}$ is equipped with a $\mathbb{G}(\Xi)$-graded $C^{*}$-algebra structure with components $\mathscr{L}(E)$.

In the rest of this section we prove Theorem 2.9. Note that

[TABLE]

Indeed, it suffices to show that $\mu\oast\nu=\nu\oast\mu$ if $\mu\in M(E)$ and $\nu\in M(F)$. If $G\in\mathbb{G}_{\mathrm{s}}(\Xi)$ with $E+F\subset G$ and $f\in C_{0}(G)$ then by (2.2)

[TABLE]

because $\sigma(\xi,\eta)=0$ if $\xi\in E,\eta\in F$ and we have the same formula for $\nu\oast\mu$.

Lemma 2.11**.**

If $\Xi$ is finite dimensional, $\mu=\sum_{E\in\mathbb{G}(\Xi)}\mu(E)\in\mathring{\mathscr{L}}$, and $\xi\in\Xi$, then

[TABLE]

Proof.

We have

[TABLE]

Indeed, (2.10) ensures this for $\mu\in M(\Xi)$ and then the relation extends to any $\mu\in\mathbb{S}^{\prime}(\Xi)$ by continuity and density of $M(\xi)$ in $\mathbb{S}^{\prime}(\Xi)$; then we use the embedding $\mathscr{M}(\Xi)\subset\mathbb{S}^{\prime}(\Xi)$.

Now let $E$ be a subspace of $\Xi$ and $\mu\in\mathscr{L}(E)$. From (2.19) and (2.9) we get

[TABLE]

On the other hand, as weak limit in $\mathbb{S}^{\prime}(\Xi)$ we have

[TABLE]

For the proof, note that, by the last relation in (2.10) and a density and continuity argument, for any $\mu,\nu\in\mathscr{M}(\Xi)$ we have

[TABLE]

hence, since $\mathscr{L}(E)$ is the norm closure of $L^{1}(E)$ in $\mathscr{M}(\Xi)$, it suffices to prove (2.23) for $\mu\in L^{1}(E)$. If $\xi\notin E^{\sigma}$ then $\xi_{\sigma}|_{E}$ is a nonzero linear form on $E$ hence by (2.21) if $\theta\in\mathbb{S}(\Xi)$

[TABLE]

tends to zero as $r\to\infty$ by the Riemann-Lebesgue lemma. This proves (2.20). ∎

We prove (1) of the Theorem 2.9, i.e. if $\mu(E)\in\mathscr{L}(E)$ $\forall E\in\mathcal{S}=$ finite, then

[TABLE]

Since there is a finite dimensional symplectic subspace of $\Xi$ containing all the subspaces $E\in\mathcal{S}$, it suffices to show this under the assumption that $\Xi$ is finite dimensional. For any $\xi\in\Xi$ we will have

[TABLE]

Fix $F\in\mathcal{S}$ and consider the spaces $E\in\mathcal{S}$ such that $E\not\subset F$. For such $E$ we have $F^{\sigma}\not\subset E^{\sigma}$ hence $E^{\sigma}\cap F^{\sigma}$ is a strict subspace of $F^{\sigma}$, so we may choose $\xi\in F^{\sigma}$ which does not belong to any of these subspaces, hence $\xi\notin E^{\sigma}$ if $E\not\subset F$. On the other hand, if $E\subset F$ then $\xi\in F^{\sigma}\subset E^{\sigma}$. Thus from (2.22) and (2.23) we get

[TABLE]

If $F$ is minimal in $\mathcal{S}$ we get $\mu_{F}=0$. Then if $\mathcal{S}_{1}$ is the set of $E\in\mathcal{S}$ which are not minimal, we get $\sum_{E\in\mathcal{S}_{1}}\mu(E)=0$. By repeating the above argument for $\mathcal{S}_{1}$ we get $\mu_{F}=0$ for all $F$ minimal in $\mathcal{S}_{1}$, etc. This proves part (1) of the theorem

For the proof of (2) of Theorem 2.9 we may assume $\Xi$ finite dimensional, cf. (1.10). If $\mu\in M(E)$ and $\nu\in M(F)$ then $\mbox{\rm supp\! }(\mu\oast\nu)\subset E+F$ because if $f\in C_{0}(\Xi)$ has support disjoint from $E+F$ then the right hand side of (2.2) is equal to zero. Thus $\mathscr{M}(E)\dot{\oast}\mathscr{M}(F)\subset\mathscr{M}(E+F)$, in particular we have (2.8). Alternatively, we may use $|\mu\oast\nu|\leq|\mu|\star|\nu|$, where $\star$ is the ordinary convolution operation of measures, and the following known fact: if $\mu,\nu$ are positive bounded measures on $\Xi$ then $\mbox{\rm supp\! }(\mu\star\nu)$ is included in the closure of $\mbox{\rm supp\! }\mu+\mbox{\rm supp\! }\nu$.

Now we prove that $\mu\oast\nu\in L^{1}(E+F)$ if $\mu\in L^{1}(E),\nu\in L^{1}(F)$. By the preceding comments, it suffices to prove that $\mu\star\nu\in L^{1}(E+F)$ if $\mu,\nu$ are positive measures in $L^{1}(E)$ and $L^{1}(F)$ respectively. Denote $\mu\otimes\nu$ the product measure on $E\oplus F$. This is clearly an absolutely continuous positive bounded measure on $E\oplus F$ and if we denote $S:E\oplus F\to E+F$ the sum operation $S(\xi,\eta)=\xi+\eta$ then $\mu\star\nu$ is the bounded positive measure on $E+F$ defined by $(\mu\star\nu)(A)=(\mu\otimes\nu)(S^{-1}(A))$ for any Borel set $A\subset E+F$. We have to show that this measure is absolutely continuous. But $S$ is a linear surjective map, hence if $N\subset E+F$ is of measure zero then $S^{-1}(N)$ is of measure zero. Indeed, if $G$ is a subspace of $E\oplus F$ supplementary to $\ker S$ then $S:G\to E+F$ is a linear bijective map, so $M=S^{-1}(N)\cap G$ is of measure zero and $S^{-1}(N)=G\oplus\ker S$ is also of measure zero by Fubini theorem.

Thus we have $\mathscr{L}(E)\oast\mathscr{L}(F)\subset\mathscr{L}(E+F)$ and for the proof of (3) of Theorem 2.9 it remains to show that $\mathscr{L}(E)\oast\mathscr{L}(F)$ is dense in $\mathscr{L}(E+F)$. For this it suffices to show that the only function $f\in L^{\infty}(E+F)$ such that $\int f(\xi)(\mu\oast\nu)(\mathrm{d}\xi)=0$ for all $\mu\in L^{1}(E)$ and $\nu\in L^{1}(F)$ is $f=0$. More explicitly, this condition means

[TABLE]

where $\mathrm{d}\xi,\mathrm{d}\eta$ are the Euclidean measures associated to some scalar products on $E,F$ respectively. For $\xi_{0}\in E$, $\eta_{0}\in F$, and $r>0$ real let us take above $\mu$ equal to the characteristic function of the ball $|\xi-\xi_{0}|<r$ in $E$ divided by its volume $B_{E}(r)$ and $\nu$ the similarly defined function on $F$. Then we get

[TABLE]

By the Lebesgue differentiation theorem, the limit as $r\to 0$ of the left hand side above is equal to $\mathrm{e}^{-\frac{\mathrm{i}}{2}\sigma(\xi_{0},\eta_{0})}f(\xi_{0}+\eta_{0})$ for almost every $\xi_{0}\in E,\eta_{0}\in F$. Thus $f=0$.

This finishes the proof of (3). By (1), (2) and (2.18) $\mathscr{L}$ is a $\mathbb{G}(\Xi)$-graded $C^{*}$-algebra hence (4) is true by (1) of Lemma A.1. This finishes the proof of the theorem.

Remark 2.12**.**

From Theorems 2.9 and 2.8 it follows that for each $L\in\mathbb{P}(\Xi)$ we have an abelian $C^{*}$-subalgebra $\mathscr{L}(L)$ of $\mathscr{L}$ with $\mathscr{L}(L)\cong C_{0}(L)\simeq C_{0}(\mathbb{R})$ such that the family of subspaces $\{\mathscr{L}(L)\}_{L\in\mathbb{P}(\Xi)}$ is linearly independent and generates $\mathscr{L}$. In other terms, the linear sum $\mathscr{I}=\sum_{L\in\mathbb{P}(\Xi)}\mathscr{L}(L)\subset\mathscr{L}$ is direct and the $C^{*}$-algebra generated by $\mathscr{I}$ is $\mathscr{L}$. Thus $\mathscr{L}$ is the field $C^{*}$-algebra introduced by Kastler in [40, §6].

Remark 2.13**.**

By Theorem 2.8 the algebras $\mathscr{L}(E)$ have a rather simple structure: they are isomorphic to $C^{*}$-algebras of the form $\mathscr{A}\otimes\mathscr{B}$ with $\mathscr{A}$ an abelian $C^{*}$-algebra and $\mathscr{B}$ an elementary $C^{*}$-algebra. This and [43, Prop. 4.2] imply for example that $\mathscr{L}$ is a nuclear $C^{*}$-algebra. This has been proved before in [13, Th. 3.8] by other techniques.**

2.4. Projection morphisms

For any subset $\mathcal{S}\subset\mathbb{G}(\Xi)$ we set

[TABLE]

By Theorem 2.9 if $\mathcal{S}$ is a subsemilattice of $\mathbb{G}(\Xi)$ then $\mathring{\mathscr{L}}(\mathcal{S})$ is a $*$-subalgebra and $\mathscr{L}(\mathcal{S})$ an $\mathcal{S}$-graded $C^{*}$-subalgebra of $\mathscr{L}$. If $\mathcal{S}$ is finite then $\mathscr{L}(\mathcal{S})=\mathring{\mathscr{L}}(\mathcal{S})$ is a $C^{*}$-subalgebra of $\mathscr{L}$ if and only if $\mathcal{S}$ is a subsemilattice of $\mathbb{G}(\Xi)$, by Lemma A.1 and Theorem 2.9-(3).

The linear direct sum decomposition $\mathring{\mathscr{L}}=\sum_{E\in\mathbb{G}(\Xi)}\mathscr{L}(E)$ gives us a linear projection $\mathring{\mathcal{P}}(\mathcal{S})\colon\mathring{\mathscr{L}}\to\mathring{\mathscr{L}}(\mathcal{S})$ for any $\mathcal{S}$. Then $\mathring{\mathcal{P}}(E)=\mathring{\mathcal{P}}(\{E\}):\mathring{\mathscr{L}}\to\mathscr{L}(E)$ is the linear projection determined by the direct sum decomposition and $\mathring{\mathcal{P}}(\mathcal{S})=\sum_{E\in\mathcal{S}}\mathring{\mathcal{P}}(E)$. If $\mathring{\mathcal{P}}(\mathcal{S})$ extends to a continuous map $\mathscr{L}\to\mathscr{L}(\mathcal{S})$ we denote $\mathcal{P}(\mathcal{S})$ the extension, this will be a projection of $\mathscr{L}$ onto the subspace $\mathscr{L}(\mathcal{S})$.

Any subspace $E\subset\Xi$ determines three subsemilattices of interest defined as in (A.1) and the corresponding $C^{*}$-subalgebras will be denoted $\mathscr{L}_{\scriptscriptstyle E},\mathscr{L}^{\prime}_{\scriptscriptstyle E},\mathscr{L}_{\scriptscriptstyle\supset E}$. Thus

[TABLE]

Clearly $\mathscr{L}_{\scriptscriptstyle E}$ is a unital $C^{*}$-subalgebra of $\mathscr{L}$ graded by $\mathbb{G}(E)$ and $\mathscr{L}^{\prime}_{\scriptscriptstyle E}$ and $\mathscr{L}_{\scriptscriptstyle\supset E}$ are ideals graded by the obvious subsemilattices. From Theorem A.2 and Proposition A.3 we get:

1. (1)

$\mathscr{L}=\mathscr{L}_{\scriptscriptstyle E}+\mathscr{L}^{\prime}_{\scriptscriptstyle E}$ and $\mathscr{L}_{\scriptscriptstyle E}\cap\mathscr{L}^{\prime}_{\scriptscriptstyle E}=0$. 2. (2)

The projection $\mathcal{P}_{\scriptscriptstyle E}\colon\mathscr{L}\to\mathscr{L}_{\scriptscriptstyle E}$ determined by the preceding direct sum decomposition is a morphism such that $\mu={\textstyle\sum_{{\scriptscriptstyle F}}}\mu(F)\in\mathring{\mathscr{L}}\Rightarrow\mathcal{P}_{\scriptscriptstyle E}\mu={\textstyle\sum_{\scriptscriptstyle F\subset E}}\mu(F)$. 3. (3)

$\mathscr{L}_{\scriptscriptstyle E\cap F}=\mathscr{L}_{\scriptscriptstyle E}\cap\mathscr{L}_{\scriptscriptstyle F}\quad\text{and}\quad\mathcal{P}_{\scriptscriptstyle E\cap F}=\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle F}=\mathcal{P}_{\scriptscriptstyle F}\mathcal{P}_{\scriptscriptstyle E}$.

Thus $\mathcal{P}_{E}$ is a projection morphism of $\mathscr{L}$ onto its subalgebra $\mathscr{L}_{E}$.

Remark 2.14**.**

In particular, $\mathscr{L}$ has many ideals, fact established in [13] by different techniques. $\mathscr{L}$ has many other ideals, e.g. $\mathscr{L}(\mathcal{J})$ is an ideal if $\mathcal{J}\subset\mathbb{G}(\Xi)$ satisfies $E\in\mathcal{J},F\supset E\Rightarrow F\in\mathcal{J}$. If $\mathcal{J}_{k}=\{E\in\mathbb{G}(\Xi)\mid\dim E\geq k\}$ we get a sequence of ideals $\mathscr{L}_{(k)}\doteq\mathscr{L}(\mathcal{J}_{k})$ such that $\mathscr{L}_{(0)}=\mathscr{L}$ and $\mathscr{L}_{(k)}\supset\mathscr{L}_{(k+1)}$.

Remark 2.15**.**

If $E=0$ then $\mathscr{L}_{0}=\mathscr{L}(0)=\mathbb{C}$ and $\mathcal{P}_{0}\colon\mathscr{L}\to\mathbb{C}$ is a projection morphism and a trace because $\mathcal{P}_{0}(\mu\oast\nu)=\mathcal{P}_{0}(\mu)\mathcal{P}_{0}(\nu)=\mathcal{P}_{0}(\nu\oast\mu)$.

Let $\mathscr{L}_{E}^{\mathrm{com}}=\{\nu\in\mathscr{L}\mid\nu\oast\mu=\mu\oast\nu\ \forall\mu\in\mathscr{L}_{E}\}$. From Theorem 2.36 we get:

Proposition 2.16**.**

If $\Xi$ is finite dimensional then for any subspace $E\subset\Xi$:

[TABLE]

We mention a consequence of Lemma 2.11 which says that, if $\Xi$ is finite dimensional, the projection morphism $\mathcal{P}_{E}$ associated to a hyperplane $E$ may be thought as a “translation at infinity” in a direction $\sigma$-orthogonal to $E$. In Proposition 2.34 we prove a similar result in a representation $\mathscr{F}$ of $\mathscr{L}$ for $\Xi$ of any dimension. If $\xi\in\Xi$ we set $\xi^{\sigma}\doteq(\mathbb{R}\xi)^{\sigma}$, hence we have $\xi\in F^{\sigma}\Leftrightarrow F\subset\xi^{\sigma}$. Any hyperplane $E\subset\Xi$ is of the form $E=\xi^{\sigma}$ with $\xi\in E^{\sigma}\!\setminus\!\{0\}$.

Proposition 2.17**.**

If $\Xi$ is finite dimensional, $\xi\in\Xi\setminus\!\{0\}$, and $E=\xi^{\sigma}$, then $\forall\mu\in\mathscr{L}$

[TABLE]

We now extend this interpretation of $\mathcal{P}_{E}$ to the case when $E$ is not a hyperplane. Note that by Lemma A.1-(3) any $\mu\in\mathscr{L}$ belongs to some $\mathscr{L}(\mathcal{S})$ with $\mathcal{S}$ a countable subsemilattice.

Proposition 2.18**.**

Assume $\Xi$ finite dimensional and let $E\in\mathbb{G}(\Xi)$ and $\mathcal{S}\subset\mathbb{G}(\Xi)$ a countable subsemilattice. Then there is $\xi\in E^{\sigma}$ such that $\xi\notin F^{\sigma}$ if $F\in\mathcal{S},F\not\subset E$, and for any such $\xi$ and any $\mu\in\mathscr{L}(\mathcal{S})$

[TABLE]

Proof.

If $F\not\subset E$ then $E^{\sigma}\not\subset F^{\sigma}$ hence $E^{\sigma}\cap F^{\sigma}$ is a strict subspace of $E^{\sigma}$ which cannot be a countable union of strict subspaces so $\xi$ exists and is not zero. Then it suffices to consider $\mu\in\mathring{\mathscr{L}}(\mathcal{S})$ hence $\mu=\sum_{F\in\mathcal{S}}\mu(F)$ with only a finite number of nonzero terms hence $\mathcal{P}_{E}\mu=\sum_{F\subset E}\mu(F)$. On the other hand

[TABLE]

and if $r\to\infty$ then each term in the last sum tends to zero by Lemma 2.11. ∎

2.5. Field observables

Our purpose here is to give a description of the field algebra in terms of fields thus making the link with the Buchholz-Grundling approach. We define the field operators at an abstract level as observables affiliated to $\mathscr{L}$ and express the components $\mathscr{L}(E)$ and the subalgebras $\mathscr{L}_{E}$ in terms of the fields.

2.5.1. Fourier algebra and observables

Observables affiliated to a $C^{*}$-algebra $\mathscr{C}$ may be described by using $*$-subalgebras $\mathcal{A}$ of $C_{0}(\mathbb{R})$ such that any morphism $\mathcal{A}\to\mathscr{C}$ extends uniquely to a morphism on $C_{0}(\mathbb{R})$. For example, the algebra of continuous rational functions of degree $<0$ has this property and gives the description of observables in terms of self-adjoint resolvents. Here we show that the Fourier algebra allows one to define observables as generators of unitary groups in the multiplier algebra of $\mathscr{C}$.

More generally, if $X$ is a locally compact abelian group, we construct $X$-valued observables affiliated to $\mathscr{C}$, i.e. morphisms $\phi:\mathcal{A}(X)\to\mathscr{C}$, by using the Fourier algebra of $X$. Let $X^{*}$ be the space of characters of $X$ equipped with a Haar measure $\mathrm{d}\chi$. We define the Fourier transformation by the condition $\theta(x)=\int_{X^{*}}\chi(x)\hat{\theta}(\chi)\mathrm{d}\chi$. The set of Fourier transforms of integrable functions on $X^{*}$ is a dense stable under conjugation and translations subalgebra of $C_{0}(X)$ which, when equipped with the norm $\|\theta\|=\|\hat{\theta}\|_{L^{1}(X^{*})}$, becomes a Banach $*$-algebra $\mathcal{A}(X)$ continuously embedded into $C_{0}(X)$ called Fourier algebra of $X$.

Proposition 2.19**.**

If $\mathscr{C}$ is a $C^{*}$-algebra and $\phi:\mathcal{A}(X)\to\mathscr{C}$ is a morphism then $\phi$ has a unique extension to a morphism $C_{0}(X)\to\mathscr{C}$.

Proof.

Since any homomorphism from a Banach algebra into a $C^{*}$-algebra is continuous [18, Prop. 4.2], the morphism $\phi:\mathcal{A}(X)\to\mathscr{C}$ is continuous. We may assume that $\mathscr{C}$ is commutative. If $Y$ is the spectrum of $\mathscr{C}$ then $Y$ is a locally compact space, $\mathscr{C}\simeq C_{0}(Y)$, so $\phi:\mathcal{A}(X)\to C_{0}(Y)$ is a continuous morphism. If $y\in Y$ then the map $\theta\mapsto\phi(\theta)(y)$ is clearly a character of $\mathcal{A}(X)$. By [37, Cor. 23.7] the spectrum of $\mathcal{A}(X)$ can be identified with $X$ with the help of the evaluation characters $\theta\mapsto\theta(x)$, hence there is a unique $x=u(y)\in X$ such that $\phi(\theta)(y)=\theta(u(y))$. Thus we get a function $u:Y\to X$ such that $\phi(\theta)=\theta\circ u$ hence $\|\phi(\theta)\|\leq\sup_{x}|\theta(x)|=\|\theta\|_{C_{0}(X)}$. ∎

Assume that $\mathscr{I}$ is an essential ideal of a unital $C^{*}$-algebra $\mathscr{C}$ and equip $\mathscr{C}$ with the strict topology defined by the family of seminorms $\|S\|_{J}=\|SJ\|+\|JS\|$ where $J$ runs over $\mathscr{I}$. If $U=\{U_{t}\}_{t\in\mathbb{R}}$ is a one parameter group of unitary elements of $\mathscr{C}$ then the continuity of the map $t\mapsto U_{t}\in\mathscr{C}$ in the strict topology is equivalent to the norm continuity of $t\mapsto U_{t}J$ for any $J\in\mathscr{I}$; if this is satisfied we say that $U$ is strictly continuous. Then, by Proposition 2.19, we may define an observable $A$ affiliated to $\mathscr{C}$ by requiring $\theta(A)=\int\hat{\theta}(t)U_{t}\mathrm{d}t$ if $\theta\in\mathcal{A}(\mathbb{R})$ the integral being taken in the strict topology. We say that $A$ is the infinitesimal generator of $U$ and we write $U_{t}=\mathrm{e}^{\mathrm{i}tA}$. We reformulate these remarks as a proposition and give an alternative proof in terms of double centralizers.

Proposition 2.20**.**

Assume that $\mathscr{I}$ is an essential ideal of the unital $C^{*}$-algebra $\mathscr{C}$ and $\{U_{t}\}_{t\in\mathbb{R}}$ is a group of unitary elements of $\mathscr{C}$ such that $t\mapsto U_{t}J$ is norm continuous for any $J\in\mathscr{I}$. Then $\{U_{t}\}_{t\in\mathbb{R}}$ has an infinitesimal generator affiliated to $\mathscr{C}$.

Proof.

For each $\theta\in\mathcal{A}(\mathbb{R})$ we define continuous maps $L_{\theta},R_{\theta}:\mathscr{I}\to\mathscr{I}$ by

[TABLE]

Clearly the pair $(L_{\theta},R_{\theta})$ is a double centralizer of the $C^{*}$-algebra $\mathscr{I}$ [45, p. 38], e.g. $J_{1}L_{\theta}(J_{2})=R_{\theta}(J_{1})J_{2}$, hence it is an element of the multiplier algebra of $\mathscr{I}$ which is $\mathscr{C}$ because $\mathscr{I}$ is an essential ideal of $\mathscr{C}$. Thus there is an element $\theta(A)\in\mathscr{C}$, formally equal to $\int\check{\theta}(t)U_{t}\mathrm{d}t$, such that $L_{\theta}(J)=\theta(A)J$ and $R_{\theta}(J)=J\theta(A)$. It is easy to check that $\theta\mapsto\theta(A)$ is a morphism [45, p. 39] hence by Proposition 2.19 this map defines a self-adjoint operator $A$ affiliated to $\mathscr{C}$. ∎

2.5.2. Field observables

Now we go back to our framework. We need two simple facts.

Lemma 2.21**.**

If $E\in\mathbb{G}_{\mathrm{s}}(\Xi)$ then $\mathscr{L}(E)$ is an essential ideal of $\mathscr{M}(E)$.

If $E\in\mathbb{G}_{\mathrm{s}}(\Xi)$ then the strict topology on $\mathscr{M}(E)$ is that associated to $\mathscr{L}(E)$.

Lemma 2.22**.**

If $\xi\in E\in\mathbb{G}_{\mathrm{s}}(\Xi)$ then the family $\{\delta_{t\xi}\}_{t\in\mathbb{R}}$ is a strictly continuous one parameter group of unitary elements in $\mathscr{M}(E)$.

Both lemmas are easily proven. Thanks to Lemma 2.22 and Proposition 2.20 we may now define the fields as observables affiliated to the Kastler algebra $\mathscr{M}(\Xi)$.

Definition 2.23**.**

The field observable $\phi(\xi)$ at the point $\xi\in\Xi$ is the infinitesimal generator of the one parameter group $\{\delta_{t\xi}\}_{t\in\mathbb{R}}$ of unitary elements in $\mathscr{M}(\Xi)$.

$\phi(\xi)$ is affiliated to any $C^{*}$-algebra $\mathscr{M}(E)$ such that $\xi\in E\in\mathbb{G}_{\mathrm{s}}(\Xi)$ and a priori it could depend on $E$, so should be denoted $\phi_{E}(\xi)$. But if $E\subset F\in\mathbb{G}_{\mathrm{s}}(\Xi)$ then clearly $\phi_{F}(\xi)=\phi_{E}(\xi)$ and then by (1.10) $\phi_{E}(\xi)$ is independent of $E$ so we may denote it $\phi(\xi)$. This is the observable affiliated to $\mathscr{M}(\Xi)$ defined by $\theta(\phi(\xi))=\int_{\mathbb{R}}\hat{\theta}(t)\delta_{t\xi}\mathrm{d}t$ for all $\theta\in\mathcal{A}(\mathbb{R})$.

Theorem 2.24**.**

(1)* The $C^{*}$-algebra generated by $\phi(\xi)$ is $C^{*}(\phi(\xi))=\mathscr{L}(\mathbb{R}\xi)$.

(2) If $\{\xi_{1},\dots,\xi_{n}\}$ is a generating set for the subspace $E\in\mathbb{G}(\Xi)$ then*

[TABLE]

(3)* If $E\subset\Xi$ is an arbitrary subspace then*

[TABLE]

Proof.

The assertion (1) is trivial if $\xi=0$, so let $\xi\neq 0$. If $\theta\in\mathcal{A}(\mathbb{R})$ then $\theta(\phi(\xi))=\int\hat{\theta}(t)\delta_{t\xi}\mathrm{d}t$ is a measure on the line $\mathbb{R}\xi$ acting as follows: $\int f\theta(\phi(\xi))=\int_{\mathbb{R}}\hat{\theta}(t)f(t\xi)\mathrm{d}t$ if $f\in C_{0}(\mathbb{R}\xi)$. Thus $\theta(\phi(\xi))$ is absolutely continuous, i.e. belongs to $L^{1}(\mathbb{R}\xi)\subset\mathscr{L}(\mathbb{R}\xi)$. It is clear that the $C^{*}$-algebra generated by $\phi(\xi)$ is the closure in $\mathscr{L}(\mathbb{R}\xi)$ of the set of elements $\theta(\phi(\xi))$ with $\theta\in\mathcal{A}(\mathbb{R})$, or this set is clearly dense in $L^{1}(\mathbb{R}\xi)$ which in turn is dense in $\mathscr{L}(\mathbb{R}\xi)$. This finishes the proof of (1). Then (1) and (3) of Theorem 2.9 imply (2). Finally, (3) is an easy consequence of the definition (2.27). ∎

We adopt (2.32) as definition of $\mathscr{L}_{E}$ for any subset $E\subset\Xi$. Clearly $E\subset F$ $\Rightarrow$ $\mathscr{L}_{E}\subset\mathscr{L}_{F}$. We have $\mathscr{L}_{\emptyset}=0$, $\mathscr{L}_{0}=\mathbb{C}$ (because $\phi(0)=0$), and if $E=\{\xi\}$ with $\xi\neq 0$ then $\mathscr{L}_{E}=C^{*}(\phi(\xi))$ which is not unital. Below we say that two vectors are collinear if they generate the same subspace; so [math] is not colinear with any nonzero vector.

Proposition 2.25**.**

Let $E\subset\Xi$ a finite set and $\mathcal{E}$ the set of linear subspaces generated by the subsets of $E$; then $\mathscr{L}_{E}=\sum_{F\in\mathcal{E}}\mathscr{L}(F)$. If $\xi\in\Xi$ then $\phi(\xi)$ is affiliated to $\mathscr{L}_{E}$ if and only if $\xi$ is collinear with some $\eta\in E$.

Proof.

By (4) of Theorem 2.9 the sum $\sum_{F\in\mathcal{E}}\mathscr{L}(F)$ is a closed linear subspace of $\mathscr{L}$. If $F^{\prime},F^{\prime\prime}\in\mathcal{E}$ are generated by the subsets $f^{\prime},f^{\prime\prime}$ of $E$ then $F^{\prime}+F^{\prime\prime}$ is generated by the subset $f^{\prime}\cup f^{\prime\prime}$ hence belongs to $\mathcal{E}$ and $\sum_{F\in\mathcal{E}}\mathscr{L}(F)$ is a $C^{*}$-algebra by (3) of Theorem 2.9. $\mathscr{L}_{E}$ is the $C^{*}$-algebra generated by the operators $\phi(\xi)$ with $\xi\in E$ and if $\xi_{1},\dots,\xi_{n}\in E$ generate the subspace $F$ and $u_{1},\dots,u_{n}\in C_{0}(\mathbb{R})$ then $u_{1}(\phi(\xi_{1}))\oast\dots\oast u_{n}(\phi(\xi_{n}))$ belongs to $\mathscr{L}(F)$ by (2.31) hence $\mathscr{L}_{E}\subset\sum_{F\in\mathcal{E}}\mathscr{L}(F)$. We have equality here because $\mathscr{L}(F)\subset\mathscr{L}_{E}$ by the same argument. The last assertion follows from Theorem 2.9-(1). ∎

Remark 2.26**.**

In particular, if $\xi,\eta\in\Xi$ are linearly independent and $\zeta\in\Xi$ is not collinear with any of them, then the operator $\phi(\zeta)$ is not affiliated to $C^{*}(\phi(\xi),\phi(\eta))$. Note also that if $\dim E>1$ then $\mathscr{L}_{E}$ is not generated by a finite set of operators $\phi(\xi)$ hence the definitions (1.3) and (3.32) in [33] are wrong, but this does not play any role in later arguments there, it suffices to change the quoted relations in $\Phi_{E}=C^{*}(\phi(\xi)\mid\xi\in E)$. **

2.6. Field algebra in a representation

Here we describe Hilbert space realizations of the abstract field algebra $\mathscr{L}$ obtained via representations $W$ of $\Xi$. We define them by transporting $\mathscr{L}$ with the help of the $C^{*}$-isomorphism $W\colon\mathscr{M}(\Xi)\to\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{K}}(\Xi)$.

In Definition 2.23 we introduced the “abstract” field operator $\phi(\xi)$ as an observable affiliated to $\mathscr{M}(\Xi)$. On the other hand, in §1.3 we defined the field operator $\phi_{\scriptscriptstyle{W}}(\xi)$ as a self-adjoint operator on the Hilbert space $\mathcal{H}$. These two operators are related by the algebraic representation $W$ of Proposition 2.2: indeed, if we take $\mu=\delta_{t\xi}$ in (2.12) we get $W(\delta_{t\xi})=W(t\xi)$ which implies $W(\phi(\xi))=\phi_{\scriptscriptstyle{W}}(\xi)$. Thus the image of the “abstract” field operator $\phi(\xi)$ through the $C^{*}$-algebra representation $W$ is the field operator $\phi_{\scriptscriptstyle{W}}(\xi)$. If there is no risk of confusion, we identify these operators $\phi_{\scriptscriptstyle{W}}(\xi)\equiv\phi(\xi)$.

In the next definition we use Proposition 2.2 and Theorem 2.24 which gives a description à la Buchholz-Grundling of the field algebra and its components in the representation $W$.

Definition 2.27**.**

*Let $W$ be a representation of $\Xi$ on the Hilbert space $\mathcal{H}$.

(1) The field $C^{*}$-algebra $\mathscr{F}$ of $\Xi$ in the representation $W$ is the norm closure of the set of operators $W(\mu)$ with $\mu\in L^{1}(E)$ for some $E\in\mathbb{G}(\Xi)$. We have $\mathscr{F}=C^{*}(\phi(\xi)\mid\xi\in\Xi)$.

(2) If $E\subset\Xi$ is any linear subspace then $\mathscr{F}_{\scriptscriptstyle E}$ is the norm closure of the set of operators $W(\mu)$ with $\mu\in L^{1}(F)$ for some $F\in\mathbb{G}(E)$. We have $\mathscr{F}_{\scriptscriptstyle E}=C^{*}(\phi(\xi)\mid\xi\in E)$.

(3) If $E\in\mathbb{G}(\Xi)$ then the norm closure of the set of operators $W(\mu)$ with $\mu\in L^{1}(E)$ is a $C^{*}$-algebra $\mathscr{F}(E)$. If $\{\xi_{1},\dots,\xi_{n}\}$ is a generating set for $E$ then*

[TABLE]

(2.33) follows from Theorem 2.24 and implies the last assertions of (1) and (2). As in the purely algebraic framework we call the algebras $\mathscr{F}(E)$ components of $\mathscr{F}$. Clearly

[TABLE]

Remark 2.28**.**

The spaces $\mathscr{F},\mathscr{F}_{\scriptscriptstyle E},\mathscr{F}(E)$ from Definition 2.27 depend of $\Xi$ and $W$ but in general we do not specify this explicitly unless this is necessary and then we use the notations $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi},\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}_{\scriptscriptstyle E},\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}(E)$ or just $\mathscr{F}^{\scriptscriptstyle\Xi},\mathscr{F}^{\scriptscriptstyle\Xi}_{\scriptscriptstyle E},\mathscr{F}^{\scriptscriptstyle\Xi}(E)$. The last three simpler notations are justified by Proposition 2.30. If $\Xi$ is clear from the context we set $\mathscr{F}=\mathscr{F}^{\scriptscriptstyle\Xi}$, etc.

Remark 2.29**.**

The restriction of $W$ to a symplectic subspace $E\subset\Xi$ is a representation of $E$ on $\mathcal{H}$ still denoted $W$. If $F$ is a symplectic subspace and $E\subset F$ then $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle E}\subset\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle F}$ and (1.10) implies $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}=\overline{\cup}_{E\in\mathbb{G}_{s}(\Xi)}\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle E}$, where $\overline{\cup}$ denotes the closure of the union.**

Below, the relation $\Phi(\phi_{1}(\xi))=\phi_{2}(\xi)$ means \Phi\big{(}u(\phi_{1}(\xi))\big{)}=u(\phi_{2}(\xi)) $\forall u\in C_{0}(\mathbb{R})$ and is equivalent to $\Phi(R_{1}(\xi))=R_{2}(\xi)$ with $R_{k}(\xi)=(\phi_{k}(\xi)-\mathrm{i})^{-1}$.

Proposition 2.30**.**

The algebras $\prescript{{\scriptscriptstyle{W}}_{\!1}\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}$ and $\prescript{{\scriptscriptstyle{W}}_{\!2}\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}$ associated to two representations $W_{1},W_{2}$ of $\Xi$ are canonically isomorphic in the following sense: if $\phi_{k}(\xi)$ are the field operators in the representation $W_{k}$, then there is a unique morphism $\Phi\colon\prescript{{\scriptscriptstyle{W}}_{\!1}\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}\to\prescript{{\scriptscriptstyle{W}}_{\!2}\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}$ such that $\Phi(\phi_{1}(\xi))=\phi_{2}(\xi)$ for all $\xi\in\Xi$; this morphism is an isomorphism.

Proof.

Assume first $\Xi$ finite dimensional and let $W_{0}:\Xi\to U(\mathcal{H}_{0})$ an irreducible representation and $W:\Xi\to U(\mathcal{H})$ an arbitrary representation. By the Stone-Von Neumann theorem and with the notations of Remark 2.1 we have

[TABLE]

for any $\xi_{1},\dots,\xi_{n}\in\Xi$ and $u_{1},\dots,u_{n}\in C_{0}(\mathbb{R})$. The map $A\mapsto V^{-1}(A\otimes 1_{\mathcal{K}})V$ is a morphism $B(\mathcal{H}_{0})\to B(\mathcal{H})$ and, by the preceding relation, its restriction to $\prescript{{\scriptscriptstyle{W}}_{\!0}\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}$ is an isomorphism $\Phi\colon\prescript{{\scriptscriptstyle{W}}_{\!0}\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}\to\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}$ which satisfies $\Phi(\phi_{0}(\xi))=\phi(\xi)$ for all $\xi\in\Xi$ and is uniquely determined by this relation. If $\Xi$ is infinite dimensional for each $E\in\mathbb{G}_{s}(\Xi)$ we have a canonical isomorphism $\Phi_{E}\colon\prescript{{\scriptscriptstyle{W}}_{\!1}\!\!}{}{\mathscr{F}}^{\scriptscriptstyle E}\to\prescript{{\scriptscriptstyle{W}}_{\!2}\!\!}{}{\mathscr{F}}^{\scriptscriptstyle E}$ and by uniqueness $\Phi_{F}|\mathscr{F}_{E}^{{\scriptscriptstyle{W}}_{1}}=\Phi_{E}$ if $E,F\in\mathbb{G}_{s}(\Xi)$ and $E\subset F$. Then we use Remark 2.29. ∎

Remark 2.31**.**

If $\dim\Xi<\infty$ then by the Stone-Von Neumann theorem $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}\cong\prescript{{\scriptscriptstyle{W}}_{\!0}\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}\otimes 1_{\mathcal{K}}$ and $\prescript{\scriptscriptstyle W\!\!\!}{}{\mathscr{F}}^{\scriptscriptstyle\Xi}(\Xi)\cong K(\mathcal{H}_{0})\otimes 1_{\mathcal{K}}$ with the notations of Remark 2.1.

In the next proposition we describe some simple properties of the algebras $\mathscr{F}(E)$.

Proposition 2.32**.**

Let $E,F$ be finite dimensional subspaces of $\Xi$.

1. (a)

$\mathscr{F}(E)$* is a non-degenerate $C^{*}$-subalgebra of $B(\mathcal{H})$.* 2. (b)

$\mathscr{F}(0)=\mathbb{C}$* and this is the only $\mathscr{F}(E)$ which is unital.* 3. (c)

$\mathscr{F}(E)$* is abelian if and only if $E$ is isotropic and then $\mathscr{F}(E)\cong C_{0}(E^{*})$.* 4. (d)

If $S\in\mathscr{F}(E)$ and $\xi\in E^{\sigma}$ then $SW(\xi)=W(\xi)S$. 5. (e)

If $S\in\mathscr{F}(E)$ and $T\in\mathscr{F}_{E^{\sigma}}$ then $ST=TS$.

Proof.

Let $\lambda_{E}$ be a Lebesgue measure on $E$ and $\rho$ an integrable function on $E$ with $\int_{E}\rho\lambda_{E}=1$. Then for $\xi\in E$ and $\varepsilon>0$ consider the function $\rho^{\varepsilon}_{\xi}(\eta)=\varepsilon^{-n}\rho((\eta-\xi)/\varepsilon)$, where $n=\dim E$, and let $\mu^{\varepsilon}_{\xi}(\mathrm{d}\eta)=\rho^{\varepsilon}_{\xi}(\eta)\lambda_{E}(\mathrm{d}\eta)$. Then we have

[TABLE]

hence

[TABLE]

For example $\operatorname*{s-lim}_{\varepsilon\to 0}W(\mu_{0}^{\varepsilon})=1$, which implies property (a). The first assertion of (b) is obvious and the second follows from $\mathscr{F}(E)\cap\mathscr{F}(F)=0$ if $E\neq F$. If $\mathscr{F}(E)$ is abelian then $W(\mu)W(\nu)=W(\nu)W(\mu)$ for all $\mu,\nu\in L^{1}(E)$. By taking $\mu=\mu^{\varepsilon}_{\xi}$ and $\nu=\mu^{\varepsilon}_{\eta}$ with $\xi,\eta\in E$ and making $\varepsilon\to 0$ we get $W(\xi)W(\eta)=W(\eta)W(\xi)$ and thus $(\mathrm{e}^{-\mathrm{i}\sigma(\xi,\eta)}-1)W(\eta)W(\xi)=0$ hence $\mathrm{e}^{-\mathrm{i}\sigma(\xi,\eta)}=1$ for all $\xi,\eta\in E$, so $E$ is isotropic. The converse assertion is obvious and the isomorphism with $C_{0}(E^{*})$ is discussed in Theorem 2.8. The assertions (d) and (e) are obvious. ∎

Clearly Theorem 2.9 implies Theorem 1.7 and Theorem 1.10 is a consequence of Theorem A.2 and Proposition A.3. We now show that certain projections $\mathcal{P}_{\scriptscriptstyle E}$ may be thought as “translations at infinity”.

Lemma 2.33**.**

If $F\in\mathbb{G}(\Xi)$, $\xi\notin F^{\sigma}$, and $T\in\mathscr{F}(F)$ then $\operatorname*{s-lim}_{r\to\infty}TW(r\xi)=0$

Proof.

We have to prove $\lim_{r\to\infty}\|TW(r\xi)f\|=0$ $\forall f\in\mathcal{H}$. We may assume $\Xi$ finite dimensional, if not we replace it by a finite dimensional symplectic subspace of $\Xi$ containing $F$ and $\xi$. Then it suffices to consider $T=W(\mu)$ with $\mu\in L^{1}(F)$. If $\nu=\mu^{*}\oast\mu$ then

[TABLE]

and from (2.10) we have

[TABLE]

with $\nu\in L^{1}(E)$, hence

[TABLE]

The function $\theta\doteq\sigma(\xi,\cdot)|F$ is a nonzero linear form on $F$ and the integral above is of the form $\int_{F}\mathrm{e}^{\mathrm{i}r\theta(\eta)}u(\eta)\mathrm{d}\eta$ with $u\in L^{1}(F)$ and if $r\to\infty$ such an integral tends to zero by the Riemann-Lebesgue lemma, which finishes the proof ∎

Proposition 2.34**.**

Let $\xi\in\Xi\setminus\{0\}$ and $E=\xi^{\sigma}$. Then for any $T\in\mathscr{F}$

[TABLE]

Proof.

$\mathcal{P}_{\scriptscriptstyle E}$ is continuous so it suffices to prove (2.36) for $T\in\mathring{\mathscr{F}}$ and for this it suffices to consider the case $T\in\mathscr{F}(F)$ for some $F\in\mathbb{G}(\Xi)$. By (d) of Proposition 2.32 for $F\subset E$ we have $TW(r\xi)=W(r\xi)T$ $\forall r\in\mathbb{R}$ hence $\operatorname*{s-lim}_{r\to\infty}W(r\xi)^{*}TW(r\xi)=S$. If $F\not\subset E=\xi^{\sigma}$ then $\xi\notin F^{\sigma}$ hence from Lemma 2.33 we get $\operatorname*{s-lim}_{r\to\infty}W(r\xi)^{*}TW(r\xi)=0$. ∎

The following proposition is proved by an argument similar to that of Proposition 2.18; the last assertion is a consequence of Lemma A.1-(3).

Proposition 2.35**.**

Let $\Xi$ be finite dimensional, $E\in\mathbb{G}(\Xi)$, and $\mathcal{S}\subset\mathbb{G}(\Xi)$ a countable subsemilattice. Then $\exists\,\xi\in E^{\sigma}$ such that $\xi\notin F^{\sigma}$ if $F\in\mathcal{S},F\not\subset E$, and for any such $\xi$

[TABLE]

In particular, for any $T\in\mathscr{F}$ and $E\in\mathbb{G}(\Xi)$ there is $\xi\in E^{\sigma}$ such that

[TABLE]

The next result improves for finite dimensional $\Xi$ the statements (d) and (e) of Proposition 2.32. If $\mathcal{A}\subset\mathscr{F}$ then $\mathcal{A}^{\mathrm{com}}\doteq\{A\in\mathscr{F}\mid[A,B]=0\ \forall B\in\mathcal{A}\}$ is its commutant in $\mathscr{F}$.

Theorem 2.36**.**

If $\Xi$ is finite dimensional then for any subspace $E\subset\Xi$ we have

[TABLE]

In particular $\mathscr{F}_{\scriptscriptstyle E}^{\mathrm{com}}\equiv(\mathscr{F}_{\scriptscriptstyle E})^{\mathrm{com}}=\mathscr{F}_{\scriptscriptstyle E^{\sigma}}$ and

[TABLE]

Proof.

If $T\in\mathscr{F}_{\scriptscriptstyle E}$ then $[T,W(\xi)]=0\ \forall\xi\in E^{\sigma}$ by Proposition 2.32-(d). Reciprocally, assume that $T$ has the last property. Then by the last assertion of Proposition 2.35 we get $\mathcal{P}_{\scriptscriptstyle E}T=T$ which is equivalent to $T\in\mathscr{F}_{\scriptscriptstyle E}$. This proves the first equality in (2.39). To prove the second, note that by Definition 2.27-(2) $\mathscr{F}_{\scriptscriptstyle E}$ is the $C^{*}$-algebras generated by the $u(\phi(\xi))$ with $u\in C_{0}(\mathbb{R})$ and $\xi\in E$ hence $T\in\mathscr{F}_{\scriptscriptstyle E}^{\mathrm{com}}$ if and only if $[T,u(\phi(\xi))]=0$ for all such $u$ and $\xi$ and this in turn is equivalent to $[T,\mathrm{e}^{\mathrm{i}\phi(\xi)}]=0$ for all $\xi\in E$. ∎

Corollary 2.37**.**

If $X$ is a Lagrangian subspace of $\Xi$ then $\mathscr{F}_{\scriptscriptstyle{X}}^{\mathrm{com}}=\mathscr{F}_{\scriptscriptstyle{X}}$.

An explicit description of the algebra $\mathscr{F}_{X}$ is given in Proposition 4.5.

We mention a fact which allows one to consider crossed products of graded $C^{*}$-subalgebras $\mathscr{F}(\mathcal{S})$ by the action of finite dimensional subspaces of $\Xi$.

Proposition 2.38**.**

(1)* If $\mathcal{S}\subset\mathbb{G}(\Xi)$ then $W(\xi)^{*}\mathscr{F}(\mathcal{S})W(\xi)=\mathscr{F}(\mathcal{S})$ $\forall\xi\in\Xi$.

(2) $\forall\xi\in\Xi$ the map $T\mapsto W^{*}(\xi)TW(\xi)$ is an automorphism of $\mathscr{F}$ and for any $T\in\mathscr{F}$ the map $\xi\mapsto W^{*}(\xi)TW(\xi)\in\mathscr{F}$ is norm continuous on finite dimensional subspaces of $\Xi$.*

Proof.

We have $W(\xi)^{*}W(\eta)W(\xi)=\mathrm{e}^{\mathrm{i}\sigma(\xi,\eta)}W(\eta)$ hence if $E\in\mathbb{G}(\Xi)$ and $\mu\in L^{1}(E)$

[TABLE]

which clearly implies (1) from which it follows that $W(\xi)^{*}\mathscr{F}W(\xi)=\mathscr{F}$ hence the first part of (2). Then the relation above implies

[TABLE]

hence $\xi\mapsto W(\xi)^{*}W(\mu)W(\xi)\in B(\mathcal{H})$ is norm continuous on finite dimensional subspaces of $\Xi$ which implies the last part of (2) by a density argument. ∎

3. Finite dimensional symplectic spaces

3.1. Intrinsic description of

$\mathscr{F}(E)$

This subsection is devoted to the proof Theorem 1.11. We use the first characterization of $\mathscr{F}(E)$ from Definition 2.27, so $\mathscr{F}(E)$ is the norm closure in $B(\mathcal{H})$ of the set of operators $W(f)=\int_{E}W(\xi)f(\xi)\mathrm{d}\xi$ with $f\in L^{1}(E)$.

We denote $\mathscr{E}(E)$ the set of operators $T$ satisfying the three conditions (1.28). Clearly $\mathscr{E}(E)$ is a $C^{*}$-algebra and we have to prove $\mathscr{F}(E)=\mathscr{E}(E)$. This is easy in two cases. First, if $E={0}$, then $E^{\sigma}=\Xi$ hence $\mathscr{E}(0)=\mathbb{C}$ because $W$ is irreducible, so this case is trivial. The second case is $E=\Xi$: then $E^{\sigma}={0}$ so $\mathscr{E}(\Xi)=K(\mathcal{H})=\mathscr{F}(\Xi)$ by the Kolmogorov-Riesz and Stone-Von Neumann theorems (or see Theorem 3.4).

It is easy to show that $\mathscr{F}(E)\subset\mathscr{E}(E)$ (cf. Proposition 5.3) so Theorem 1.11 is equivalent to

[TABLE]

The rest of this section is devoted to the proof of this fact for $E\neq 0,\Xi$.

3.1.1. ​​​

For this we need to go beyond measures and define the operators $W(\mu)$ for $\mu$ temperate distributions on $\Xi$. This Weyl pseudo-differential calculus requires some supplementary formalism, we refer to [25] for details.

Let $\mathbb{S}(\Xi)$ be the space of Schwartz test functions on $\Xi$ and $\mathbb{S}^{\prime}(\Xi)$ its dual, the space of tempered distributions. If $\mu\in\mathbb{S}^{\prime}(\Xi)$ and $f\in\mathbb{S}(\Xi)$ the value $\mu(f)$ of $\mu$ at $f$ is denoted $\langle{f},{\mu}\rangle$ but we also write this as $\int_{\Xi}f(\xi)\mu(\xi)\mathrm{d}\xi$, which is often a convenient abuse of notation. We set $\langle{f}|{\mu}\rangle=\langle{\bar{f}},{\mu}\rangle$. We have continuous linear embeddings $\mathbb{S}(\Xi)\subset L^{2}(\Xi)\subset\mathbb{S}^{\prime}(\Xi)$. By (2.4) we also have $\mathscr{F}\subset\mathscr{K}\subset\mathbb{S}^{\prime}(\Xi)$.

Although the twisted convolution product defined on $M(\Xi)$ cannot be extended to $\mathbb{S}^{\prime}(\Xi)$, it is possible to define $\mu\oast\nu$ for $\mu,\nu\in\mathbb{S}^{\prime}(\Xi)$ if one of them has compact support (as in the case of the usual convolution). And the relations (2.10) remain valid for $\mu\in\mathbb{S}^{\prime}(\Xi)$.

The map $\xi\mapsto\xi_{\sigma}=\sigma(\cdot,\xi)$ is a linear isomorphism $\Xi\to\Xi^{*}$ hence the Fourier transform of a distribution on $\Xi$ is naturally identified with a distribution on $\Xi$. More precisely, $\Xi$ is equipped with the symplectic measure $\mathrm{d}\xi$ and the Fourier measure $\,\mathchar 22\relax\mkern-12.0mud\xi=(2\pi)^{-n}\mathrm{d}\xi$; then the symplectic Fourier transform of $\mu\in L^{1}(\Xi)$ is

[TABLE]

The map $\mathcal{F}_{\!\sigma}:\mathbb{S}(\Xi)\to\mathbb{S}(\Xi)$ satisfies $\mathcal{F}_{\!\sigma}^{2}=1$ hence is a homeomorphism and extends to a self-adjoint unitary operator in $L^{2}(\Xi)$ and to a homeomorphism in $\mathbb{S}^{\prime}(\Xi)$.

For any linear subspace $E\subset\Xi$ we choose and fix a Lebesgue measure $\lambda_{E}$ or $\mathrm{d}_{E}\xi$ on $E$. We think of $\lambda_{E}$ as a distribution on $\Xi$, namely

[TABLE]

According to the conventions from §2.1.1 we embed isometrically $L^{1}(E)\subset M(E)$ hence if $\mu$ is an absolutely continuous measure on $E$ of the form $\mu(\mathrm{d}\xi)=f(\xi)\mathrm{d}_{E}\xi$ then we may use both notations $W(\mu)$ and $W(f)$.

We could continue to work with an arbitrary irreducible representation of $\Xi$ but it is simpler, and there is no loss of generality, if we assume that $W$ is the Schrödinger representation associated to a Lagrangian decomposition $\Xi=X\oplus X^{*}$ (we use the notations of §4.1). Clearly then if $f,g\in\mathbb{S}(X)$ then the map $\xi\mapsto\langle{f}|{W(\xi)g}\rangle$ belongs to ${\mathbb{S}}(\Xi)$ so if $\mu\in\mathbb{S}^{\prime}(\Xi)$ we can define $W(\mu)=\int W\mu$ as a continuous linear operator $\mathbb{S}(X)\rightarrow\mathbb{S}^{\prime}(X)$ by setting

[TABLE]

We get a map ${\mathbb{S}}^{\prime}(\Xi)\ni\mu\mapsto W(\mu)\in B(\mathbb{S}(X),\mathbb{S}^{\prime}(X))$ which is bijective [25, Th. 1.30]. We denote $T\mapsto T^{\#}$ its inverse map, hence $T=W(\mu)\Leftrightarrow\mu=T^{\#}$. Since $\mathcal{H}=L^{2}(X)$ we have $B(\mathcal{H})\subset B(\mathbb{S}(X),\mathbb{S}^{\prime}(X))$ hence to each $T\in B(\mathcal{H})$ is associated $T^{\#}\in{\mathbb{S}}^{\prime}(\Xi)$.

We mention that if $\nu\in\mathbb{S}^{\prime}(\Xi)$ is compactly supported then $W(\nu)$ leaves $\mathbb{S}(X)$ invariant and extends to a continuous operator on $\mathbb{S}^{\prime}(X)$. And then the relation $W(\mu\oast\nu)=W(\mu)W(\nu)$ remains valid for any $\mu\in\mathbb{S}^{\prime}(\Xi)$.

3.1.2. ​​​

We now start the proof of (3.1). First note the following fact.

Lemma 3.1**.**

If $T\in B(\mathcal{H})$ and $[W(\xi),T]=0$ for all $\xi\in E^{\sigma}$ then $\mbox{\rm supp\! }T^{\#}\subset E$.

Proof.

Denote $T^{\#}\equiv\mu\in\mathbb{S}^{\prime}(\Xi)$. Then $T=W(\mu)$ and by (2.10) we have

[TABLE]

Since $W:{\mathbb{S}}^{\prime}(\Xi)\rightarrow B({\mathbb{S}}(X),{\mathbb{S}}^{\prime}(X))$ is bijective, we get $(\mathrm{e}^{\mathrm{i}\xi_{\sigma}}-1)\mu$ $=0$ $\forall\xi\in E^{\sigma}$. Let now $\eta^{\circ}\notin E$. Then there is a $\xi^{\circ}\in E^{\sigma}$ such that $\mathrm{e}^{\mathrm{i}\sigma(\eta^{\circ},\xi^{\circ})}\neq 1$. So there is a compact neighborhood $V$ of $\eta^{\circ}$ such that $\mathrm{e}^{\mathrm{i}\sigma(\eta,\xi^{\circ})}-1\neq 0$ for all $\eta\in V$. If $\theta\in C_{\mathrm{c}}^{\infty}(V)$ then $\varphi\doteq\theta/(\mathrm{e}^{\mathrm{i}\xi^{\circ}_{\sigma}}-1)\in C_{\mathrm{c}}^{\infty}(V)$ so by the previous computation $\theta\mu=\varphi(\mathrm{e}^{\mathrm{i}\xi^{\circ}_{\sigma}}-1)\mu=0$. It follows that $\eta^{\circ}\notin\mbox{\rm supp\! }\mu$. ∎

We construct now a regularization of $T\in B(\mathcal{H})$. Let $\theta\in L^{1}(\Xi)$ with $\int_{\Xi}\theta(\xi)\,\mathchar 22\relax\mkern-12.0mud\xi=1$, denote $\theta_{\varepsilon}(\xi)=\varepsilon^{-2n}\theta(\xi/\varepsilon)$ with $2n=\dim\Xi$, and define for $\varepsilon>0$

[TABLE]

Lemma 3.2**.**

(i)* If  $\lim_{\xi\rightarrow 0}\|\,[W(\xi),T]\,\|=0$ then $\lim_{\varepsilon\rightarrow 0}\|T_{\varepsilon}-T\|=0$.

(ii) If  $T\in{\mathscr{E}}(E)$ then $T_{\varepsilon}\in{\mathscr{E}}(E)$ for all $\varepsilon>0$.

(iii) If  $\theta\in\mathbb{S}(\Xi)$, then $T^{\#}_{\varepsilon}\equiv(T_{\varepsilon})^{\#}=\widehat{\theta\,}_{\!\varepsilon}\,T^{\#}$.*

Proof.

Clearly

[TABLE]

which tends to zero as $\varepsilon\rightarrow 0$ under the condition of (i). For (ii), notice first that $T_{\varepsilon}$ satisfies (i) of Definition 5.1 (for all $T\in B(\mathcal{H})$). Moreover, since $T\in{\mathscr{E}}(E)$, for all $\eta\in E^{\sigma}$

[TABLE]

thus (ii) of Definition 5.1 is valid for $T_{\varepsilon}$ too. Finally, if $\eta\in E$ then

[TABLE]

which tends to zero if $\eta\rightarrow 0$. Thus $T_{\varepsilon}\in{\mathscr{E}}(E)$.

Assume now $\theta\in{\mathbb{S}}(\Xi)$ and let us compute $T^{\#}_{\varepsilon}$. If $T^{\#}=\mu$ and $f,g\in\mathbb{S}(X)$, since the operators $W(\xi)$ leave $\mathbb{S}(X)$ invariant, one has (as in the proof of Lemma 3.1):

[TABLE]

For the proof of the third equality above, note that it suffices to show it for a set of $\mu$ which is dense in ${\mathbb{S}}^{\prime}(\Xi)$, e.g. for $\mu\in{\mathbb{S}}(\Xi)$; in this case the equality is easy to justify. Anyway, since $W$ is bijective we get $T^{\#}_{\varepsilon}=\widehat{\theta\,}_{\!\varepsilon}\mu$. ∎

Let us choose $\theta$ such that $\widehat{\theta\,}\in C_{\mathrm{c}}^{\infty}(\Xi)$. Then if $T\in\mathscr{E}(E)$, due to Lemma 3.1 and (ii) and (iii) of Lemma 3.2, the distribution $T^{\#}_{\varepsilon}$ has compact support included in $E$. And by (i) of Lemma 3.2 we have $\lim_{\varepsilon\to 0}T_{\varepsilon}=T$ in norm. Thus for the proof of (3.1) it suffices to show that any operator of the form $T_{\varepsilon}$ is norm limit of operators $W(\mu)$ with $\mu\in L^{1}(E)$.

3.1.3. ​​​

Thus from now on we assume that $T\in{\mathscr{E}}(E)$ and $T^{\#}$ is a distribution on $\Xi$ whose support is a compact subset of $E$. But we may simplify the problem still further as follows.

Let $\rho\in C^{\infty}_{\mathrm{c}}(E)$ with $\rho\geq 0$ and $\int_{E}\rho(\eta)\mathrm{d}_{E}\eta=1$; for $\varepsilon>0$ set $\rho_{\varepsilon}(\eta)=\varepsilon^{-m}\rho(\eta/\varepsilon)$ where $m=\dim E$. Then $W(\rho_{\varepsilon})\in{\mathscr{E}}(E)$ by Proposition 5.3 hence $W(\rho_{\varepsilon})T\in{\mathscr{E}}(E)$, and it is easy to check that $T$ is the norm limit of the operators $W(\rho_{\varepsilon})T$ (Lemma 5.5-(ii)). Thus it suffices to prove that such products are norm limit of operators $W(\mu)$ with $\mu\in L^{1}(E)$.

Let $\rho\in C^{\infty}_{\mathrm{c}}(E)$ identified with the measure $\rho\lambda_{E}$. Then

[TABLE]

where $\rho\oast T^{\#}$ makes sense because both $\rho$ and $T^{\#}$ are compactly supported distributions. Our purpose is to show

[TABLE]

and this clearly finishes the proof of the theorem.

We begin by proving that the twisted convolution $\rho\oast T^{\#}$ also has compact support contained in $E$, and we also deduce precise information about its structure. Let $E^{\prime}$ be a vector subspace of $\Xi$, supplementary to $E$. Since $T^{\#}$ is a distribution whose compact support is contained in $E$, we may use [38, Th. 2.3.5] to get a representation of $T^{\#}$ as a finite sum

[TABLE]

Here the tensor product refers to the identification $\Xi=E\times E^{\prime}$, $u_{\alpha}$ are compactly supported distributions on $E$, and $\partial_{\scriptscriptstyle E^{\prime}}^{\alpha}\delta$ are derivatives of the Dirac measure at zero in directions of $E^{\prime}$. Then by [46, Th. 6.27] we may write each $u_{\alpha}$ as a finite sum of derivatives (in the directions of $E$) of continuous functions with compact support on $E$ so $T^{\#}$ is a finite sum

[TABLE]

Lemma 3.3**.**

There is a finite family $\{w_{\alpha}\}_{|\alpha|\leq k}$ of functions in $C^{\infty}_{\mathrm{c}}(E)$ such that

[TABLE]

Proof.

We will give a representation of the form (3.7) for each term of the sum in (3.6), then we obtain (3.7) by adding these representations. We simplify the writing and set $v=v_{\alpha\beta}$ and $T^{\#}=\partial_{\scriptscriptstyle E}^{\beta}v\otimes\partial_{\scriptscriptstyle E^{\prime}}^{\alpha}\delta$ with $v\in C_{\mathrm{c}}(E)$. If $\theta\in C^{\infty}_{\mathrm{c}}(\Xi)$ then, by (2.2) and with the abuse of notation mentioned in §3.1.1, the action of the distribution $\rho\oast T^{\#}$ on $\theta$ is

[TABLE]

If $\eta\in\Xi$ we denote by $y$ and $y^{\prime}$ its components in $E$ and $E^{\prime}$ respectively. Then $T^{\#}$ may be written $T^{\#}(\eta)\equiv\partial_{y}^{\beta}v(y)\partial_{y^{\prime}}^{\alpha}\delta(y^{\prime})=v^{(\beta)}(y)\delta^{(\alpha)}(y^{\prime})$ which of course is slightly formal and must be interpreted in the sense of distributions. Anyway, the action of the distribution $T^{\#}$ on $\mathrm{e}^{\frac{i}{2}\xi_{\sigma}}\theta(\xi+\cdot\,)$ is given by

[TABLE]

for some polynomials $\varphi_{\alpha-\gamma}(\xi)$ coming from the derivatives of order $\alpha-\gamma$ of $\mathrm{e}^{\frac{\mathrm{i}}{2}\sigma(y^{\prime},\xi)}$ with respect to $y^{\prime}$ at $y^{\prime}=0$. Then we get

[TABLE]

By taking into account the relation

[TABLE]

and since $\partial_{y}^{\lambda}\left[\partial_{y^{\prime}}^{\gamma}\theta(\xi+y+y^{\prime})\right]_{y^{\prime}=0}=\partial_{\xi}^{\lambda}\left[\partial_{y^{\prime}}^{\gamma}\theta(\xi+y+y^{\prime})\right]_{y^{\prime}=0}$ we get

[TABLE]

Now it is clear that there is a finite number of functions $w_{\gamma}\in C_{\mathrm{c}}^{\infty}(E)$ such that

[TABLE]

From Lemma 3.3 we see that (3.5) holds if we may take $k=0$ in the representation (3.7) of $\rho\oast T^{\#}$. So the proof of the Theorem 1.11 is finished once we show that the operator $W(\rho\oast T^{\#})$ is bounded if and only if $k=0$. Before starting this last step of the proof, we describe a direct sum decomposition of the symplectic space $\Xi$ determined by $E$.

3.1.4. ​​​

Recall that the center of $E$ is $E^{\mathrm{c}}\doteq E\cap E^{\sigma}$ and set $\bar{E}:=E+E^{\sigma}$. A subspace is symplectic if its center reduces to [math]. Since $\Xi$ is finite dimensional, by using $E^{\sigma\sigma}=E$ it is easy to see that $E^{\mathrm{c}}=E^{\sigma\mathrm{c}}=\bar{E}^{\mathrm{c}}=\bar{E}^{\sigma}$ and $\bar{E}=E^{\mathrm{c}\sigma}=\overline{E^{\mathrm{c}}}=\overline{E^{\sigma}}$. In what follows, we shall denote by $\oplus^{\sigma}$ and $\oplus$ the symplectic (respectively vector) direct sum, and by $\perp$ the symplectic orthogonality between elements of $\Xi$. Let now $G\subset E$ such that $E=G\oplus E^{\mathrm{c}}$ and $F\subset E^{\sigma}$ such that $E^{\sigma}=F\oplus E^{\mathrm{c}}$. Then $G$* and $F$ are symplectic*: indeed, suppose $\xi\in G$ and $\xi\perp G$; then $E^{\mathrm{c}}\subset E^{\sigma}$ shows that $\xi\perp E^{\mathrm{c}}$, so $\xi\perp(G+E^{\mathrm{c}})$, thus $\xi\in E^{\sigma}$, which means that $\xi\in E\cap E^{\sigma}=E^{\mathrm{c}}$; then $G\cap E^{\mathrm{c}}=0$ shows $\xi=0$. Hence $H\equiv G\oplus^{\sigma}F$ is a symplectic subspace too, and we have $\bar{E}=H\oplus^{\sigma}E^{\mathrm{c}}$. For, if $\xi\in H\cap E^{\mathrm{c}}$ then there are $\xi_{\scriptscriptstyle G}\in G$ and $\xi_{\scriptscriptstyle F}\in F$ such that $\xi_{\scriptscriptstyle G}=\xi-\xi_{\scriptscriptstyle F}\in E^{\sigma}-E^{\sigma}=E^{\sigma}$ which shows that $\xi_{\scriptscriptstyle G}\perp G\subset E$. Since $G$ is symplectic, $\xi_{\scriptscriptstyle G}=0$ and the same holds for $\xi_{\scriptscriptstyle F}$. On the other hand, $H+E^{\mathrm{c}}=G+F+E^{\mathrm{c}}=(G+E^{\mathrm{c}})+(F+E^{\mathrm{c}})=E+E^{\sigma}=\bar{E}$. Further, remark that $H^{\sigma}$* is also a symplectic space and $E^{\mathrm{c}}$ is a Lagrangian subspace of $H^{\sigma}$*. Indeed, $E^{\mathrm{c}}\perp H$ thus $E^{\mathrm{c}}\subset H^{\sigma}$ and is isotropic. If it where not maximal, there would exist some $\xi\in H^{\sigma}\setminus E^{\mathrm{c}}$ with $\xi\perp E^{\mathrm{c}}$. Thus $\xi\perp(E^{\mathrm{c}}+H)=\bar{E}$, i.e. $\xi\in\bar{E}^{\sigma}=E^{\mathrm{c}}$, which is absurd.

Now let $K$ be a Lagrangian subspace of $H^{\sigma}$ such that $H^{\sigma}=E^{\mathrm{c}}\oplus K$ (this will be a Lagrangian decomposition of the symplectic space $H^{\sigma}$). Then $\Xi$ splits as

[TABLE]

3.1.5. ​​​

Let us go back now to the proof of the fact that $W(\rho\oast T^{\#})$ is bounded only if $k=0$. Note that we may take $E^{\prime}=F\oplus K$. Any $\xi\in\Xi$ may be uniquely written as $\xi=\eta+\zeta_{\scriptscriptstyle F}+\zeta_{\scriptscriptstyle K}$ with $\eta\in E$, $\zeta_{\scriptscriptstyle F}\in F$ and $\zeta_{\scriptscriptstyle K}\in K$. In the next formulas we abbreviate $\lambda_{E}(\mathrm{d}\eta)=\mathrm{d}\eta$, $\lambda_{F}(\mathrm{d}\zeta_{\scriptscriptstyle F})=\mathrm{d}\zeta_{\scriptscriptstyle F}$ and $\lambda_{K}(\mathrm{d}\zeta_{\scriptscriptstyle K})=\mathrm{d}\zeta_{\scriptscriptstyle K}$. Then, by (3.3) and (3.7),

[TABLE]

which can be further developed as follows:

[TABLE]

We have used the relations $F\perp K$, hence $W(\zeta_{\scriptscriptstyle F}+\zeta_{\scriptscriptstyle K})=W(\zeta_{\scriptscriptstyle F})\,W(\zeta_{\scriptscriptstyle K})$, and $F\perp E$, hence $\sigma(\eta,\zeta_{\scriptscriptstyle F})=0$. Further, we took into account that the derivatives with respect to $\zeta_{\scriptscriptstyle K}$ at $\zeta_{\scriptscriptstyle K}=0$ of $\mathrm{e}^{\frac{\mathrm{i}}{2}\sigma(\zeta_{\scriptscriptstyle K},\eta)}$ are polynomial functions of $\eta\in E$ and $w_{\alpha}\in C^{\infty}_{\mathrm{c}}(E)$, thus there are $w_{\alpha\beta}\in C_{\mathrm{c}}^{\infty}(E)$ such that the last equality be true with $W(w_{\alpha\beta})=\int_{E}W(\eta)w_{\alpha\beta}(\eta)\mathrm{d}\eta$.

Let $F=\Omega\oplus\Omega^{*}$ and $G=\Gamma\oplus\Gamma^{*}$ be some Lagrangian decompositions of the symplectic subspaces $F$ and $G$. Thus we get a decomposition

[TABLE]

each of the three parentheses being a symplectic space equipped with a Lagrangian decomposition. We also have $E=(\Gamma\oplus\Gamma^{*})\oplus^{\sigma}E^{\mathrm{c}}$. Then $[\partial_{\scriptscriptstyle F}^{\alpha}W(\zeta_{\scriptscriptstyle F})]_{\zeta_{\scriptscriptstyle F}=0}\equiv\phi_{\alpha}(q_{\scriptscriptstyle\Omega},p_{\scriptscriptstyle\Omega})$ and $[\partial_{\scriptscriptstyle K}^{\beta}W(\zeta_{\scriptscriptstyle K})]_{\zeta_{\scriptscriptstyle K}=0}\equiv\psi_{\beta}(p_{\scriptscriptstyle E^{\scriptscriptstyle c}})$ with $\phi_{\alpha}$ and $\psi_{\beta}$ polynomial functions of degrees $|\alpha|$ and $|\beta|$ respectively and $q,p$ are the position and momentum observables associated to the space in the index. Observe that the notation $p_{\scriptscriptstyle E^{\scriptscriptstyle c}}$ is justified by the identification $(E^{\mathrm{c}})^{*}=K$. Moreover, the operators $W(\mu_{\alpha\beta})$ can be written in the form $\omega_{\alpha\beta}(q_{\scriptscriptstyle\Gamma},p_{\scriptscriptstyle\Gamma},q_{\scriptscriptstyle E^{\scriptscriptstyle c}})$ for some smooth function $\omega_{\alpha\beta}$, expression which can be rigorously interpreted in terms of the Weyl calculus (this fact, however, does not play any role in what follows). Thus we have

[TABLE]

and we have to prove that if this operator is bounded then necessarily $k=0$.

From now on we may assume that we work in the Schrödinger representation associated to $X=\Omega\oplus\Lambda$, where $\Lambda=\Gamma\oplus E^{\mathrm{c}}$. Then our Hilbert space factorizes as $\mathcal{H}=L^{2}(\Omega)\otimes L^{2}(\Lambda)$ and, if we denote $\phi_{\alpha}(q_{\scriptscriptstyle\Omega},p_{\scriptscriptstyle\Omega})$ by $S_{\alpha}$ and $\sum_{\beta}\psi_{\beta}(p_{\scriptscriptstyle E^{\scriptscriptstyle c}})\omega_{\alpha\beta}(q_{\scriptscriptstyle\Gamma},p_{\scriptscriptstyle\Gamma},q_{\scriptscriptstyle E^{\scriptscriptstyle c}})$ by $T_{\alpha}$ we have $W(\rho\oast T^{\#})=\sum_{|\alpha|\leq k}S_{\alpha}\otimes T_{\alpha}$. We show first that $S_{\alpha}\in B(L^{2}(\Omega))$ for all $\alpha$; in fact they are complex multiples of the identity operator on $L^{2}(\Omega)$.

Notice that $T_{\alpha}\in B({\mathbb{S}}(\Lambda))$ and that we may assume that the family $\{T_{\alpha}\}_{|\alpha|\leq k}$ is linearly independent in $B({\mathbb{S}}(\Lambda))$. Let $\mathscr{T}$ denote the vector space of operators $L:{\mathbb{S}}(\Lambda)\rightarrow{\mathbb{S}}(\Lambda)$ of the form $L=\sum_{i}|{u_{i}}\rangle\langle{v_{i}}|$ (finite sum), with $u_{i},v_{i}\in{\mathbb{S}}(\Lambda)$. Then we may realize $\mathscr{T}$ as a space of linear forms on $B({\mathbb{S}}(\Lambda))$ by defining $B({\mathbb{S}}(\Lambda))\ni T\mapsto\hbox{\rm Tr}\,(LT)=\sum_{i}\langle{v_{i}}|{Tu_{i}}\rangle$. Notice that ${\mathscr{T}}$ is a subspace of the dual of $B({\mathbb{S}}(\Lambda))$ which separates the points (for, if $\hbox{\rm Tr}\,(LT)=\hbox{\rm Tr}\,(LT^{\prime})$ for all $L\in\mathscr{T}$, then $\langle{u,(T-T^{\prime})v}|{=}\rangle 0$ for all $u,v\in{\mathbb{S}}(\Lambda)$). It follows that we may find a finite family of operators $\{L_{\alpha}\}\in\mathscr{T}$ such that

[TABLE]

Indeed, if $\mathscr{V}$ is the finite dimensional vector subspace generated by the family $\{T_{\alpha}\}$ (which is a basis in it) we may find a basis $\{\Psi_{\alpha}\}$ in the dual space ${\mathscr{V}}^{\prime}$ such that $\Psi_{\alpha}(T_{\beta})=\delta_{\alpha\beta}$. Thus we are reduced to prove that for each $\Psi\in\mathscr{V}^{\prime}$ we may find $L\in\mathscr{T}$ such that $\Psi(T)=\hbox{\rm Tr}\,(LT)$ for all $T\in\mathscr{V}$, i.e. that the mapping ${\mathscr{T}}\ni L\mapsto\hbox{\rm Tr}\,(L\,\cdot\,)\!\mid_{\!{}_{\scriptscriptstyle\mathscr{V}}}\in{\mathscr{V}}^{\prime}$ is surjective. Equivalently, the dual mapping ${\mathscr{V}}\equiv{\mathscr{V}}^{\prime\prime}\ni T\mapsto\hbox{\rm Tr}\,(\,\cdot\,T)\in{\mathscr{T}}^{\prime}$ has to be injective. But this is true, since $\mathscr{T}$ separates points.

Let now $L_{\alpha}=\sum_{i}|{u_{\alpha i}}\rangle\langle{v_{\alpha i}}|$ with $u_{\alpha i},v_{\alpha i}\in{\mathbb{S}}(\Lambda)$ and, for $u,v\in{\mathbb{S}}(\Omega)$, let us compute

[TABLE]

Since $W(\rho\oast T^{\#})$ is supposed bounded, this shows that all $S_{\alpha}$ are bounded. But $S_{\alpha}$ are polynomials $\phi_{\alpha}(q_{\scriptscriptstyle\Omega},p_{\scriptscriptstyle\Omega})$, so these polynomials have to be of degree zero, i.e. $\phi_{\alpha}$ are complex numbers.

Now we can write $W(\rho\oast T^{\#})=1\otimes\sum_{\alpha}\phi_{\alpha}T_{\alpha}$. Thus it remains to show that if an operator of the form $B=\sum_{\beta}\psi_{\beta}(p_{\scriptscriptstyle E^{\scriptscriptstyle c}})W(\mu_{\beta})$ is bounded in $L^{2}(\Lambda)$, then the polynomials $\psi_{\beta}$ are constants. By rearranging the sum we can assume that $B=\sum_{\gamma}p_{\scriptscriptstyle E^{\scriptscriptstyle c}}^{\gamma}W(\mu_{\gamma})$ (a basis in $E^{\mathrm{c}}$ has been chosen). Let $a=\lambda\zeta$, with $\lambda\in\mathbb{R}$ and $\zeta\in E^{\mathrm{c}}$ and for any $u,v\in{\mathbb{S}}(\Lambda)$, denote $u_{a}=\exp{[\mathrm{i}\langle{q_{\scriptscriptstyle E^{\scriptscriptstyle c}}},{a}\rangle]}u$ and $v_{a}=\exp{[\mathrm{i}\langle{q_{\scriptscriptstyle E^{\scriptscriptstyle c}}},{a}\rangle]}v$. Then $|\langle{u_{a}}|{Bv_{a}}\rangle|\leq C\,\|u\|\,\|v\|$, and on the other hand

[TABLE]

By taking $\lambda\rightarrow\infty$ we see that we must have $\sum_{|\gamma|=m}\zeta^{\gamma}\langle{u}|{W(\mu_{\gamma})v}\rangle=0$ for all $\zeta\in E^{\mathrm{c}}$ if $m\geq 1$. But this implies $\langle{u}|{W(\mu_{\gamma})v}\rangle=0$ for all $|\gamma|\geq 1$ by a standard argument. This ends the proof of Theorem 1.11.

3.2. The ideal

$\mathscr{F}(\Xi)$

Recall that $L^{1}(\Xi)$ is a Banach $*$-algebra for the twisted convolution and the definition $W(f)=\int_{\Xi}W(\xi)f(\xi)\mathrm{d}\xi$ extends $W$ to an injective morphism $W\colon L^{1}(\Xi)\to B(\mathcal{H})$. Then $\mathscr{F}(\Xi)$ is the norm closure in $B(\mathcal{H})$ of the set of such $W(f)$, hence is a $C^{*}$-algebra of operators on $\mathcal{H}$ and is the smallest graded ideal of $\mathscr{F}$. By Definition 2.27, if $\{\xi_{1},\dots,\xi_{n}\}$ is a generating set for $\Xi$ then

[TABLE]

The commutant algebra $\mathrm{Com}(\Xi)$ and the multiplicity of $W$ are introduced in the Remark 2.1. We use the following convention: if $T\in B(\mathcal{H})$ and the symbol $T^{(*)}$ appears in a relation, then that relation has to be satisfied both by $T$ and by $T^{*}$. Note that the assertion (a) below is a version of the Kolmogorov-Riesz compactness criterion.

Theorem 3.4**.**

Let $W$ be a representation of a finite dimensional symplectic space $\Xi$.

• (1)

$\mathscr{F}(\Xi)\cdot\mathrm{Com}(\Xi)=\{T\in B(\mathcal{H})\mid\lim_{\xi\to 0}\|(W(\xi)-1)T^{(*)}\|=0\}$.

• (2)

$W$* is irreducible $\Leftrightarrow\mathscr{F}(\Xi)=K(\mathcal{H})\Leftrightarrow\mathscr{K}(\Xi)\supset K(\mathcal{H})$ and then*

[TABLE]

• (3)

$W$* is of finite multiplicity $\Leftrightarrow\mathscr{F}(\Xi)\subset K(\mathcal{H})\Leftrightarrow\mathscr{F}(\Xi)\cdot\mathrm{Com}(\Xi)=K(\mathcal{H})$; then*

• (a)

$\mathcal{J}\subset\mathcal{H}$* is relatively compact $\Leftrightarrow\lim_{\xi\to 0}\sup_{h\in\mathcal{K}}\|(W(\xi)-1)h\|=0$;*

• (b)

$T\in B(\mathcal{H})$* is compact $\Leftrightarrow$ $\lim_{\xi\to 0}\|(W(\xi)-1)T\|=0$.*

• (4)

$W$* is of infinite multiplicity if and only if $\mathscr{K}(\Xi)\cap K(\mathcal{H})=0$.*

Proof.

By the Stone-Von Neumann theorem we may assume $\mathcal{H}=\mathcal{H}_{0}\otimes\mathcal{K}$ with $\mathcal{K}$ finite dimensional and $W(\xi)=W_{0}(\xi)\otimes 1$ with $W_{0}$ is an irreducible representation of $\Xi$ on $\mathcal{H}_{0}$.

We first discuss (2). If $W$ is irreducible then $\mathscr{F}(\Xi)=K(\mathcal{H})$ by a simple argument in the Schrödinger representation and this implies $K(\mathcal{H})\subset\mathscr{K}(\Xi)$. On the other hand, if $W$ is not irreducible we cannot have $K(\mathcal{H})\subset\mathscr{K}(\Xi)$. Indeed, if $\dim\mathcal{K}>1$ then an operator of the form $K\otimes F$ with $K\in B(\mathcal{H}_{0})$ and $F\in B(\mathcal{K})$ of rank one is compact on $\mathcal{H}$ but does not belong to $\mathscr{K}(\Xi)$. Then (3.9) follows from [28, Corollary 3.5].

From $\mathscr{K}_{\scriptscriptstyle{W}}(\Xi)=\mathscr{K}_{{\scriptscriptstyle{W}}_{0}}(\Xi)\otimes 1_{\mathcal{K}}$ (4). And $\mathscr{F}^{\scriptscriptstyle{W}}(\Xi)=K(\mathcal{H}_{0})\otimes 1_{\mathcal{K}}$ hence $W$ is of finite multiplicity if and only if $\mathscr{F}^{\scriptscriptstyle{W}}(\Xi)\subset K(\mathcal{H})$ which proves the first assertion of (3).

Next we prove (1). It is easy to see that $\mathrm{Com}^{\scriptscriptstyle{W}}(\Xi)$ coincides with the commutant $\mathscr{F}^{\scriptscriptstyle{W}}(\Xi)^{\prime}$ of the algebra $\mathscr{F}^{\scriptscriptstyle{W}}(\Xi)$ (cf. the argument which proves (a) of Proposition 2.32). On the other hand, the commutant of $K(\mathcal{H}_{0})\otimes 1_{\mathcal{K}}$ coincides with the commutant of its weak closure, which is $B(\mathcal{H}_{0})\otimes 1_{\mathcal{K}}$, and by the commutation theorem for tensor products the commutant of the last algebra is $1_{\mathcal{H}_{0}}\otimes B(\mathcal{K})$. Thus $\mathrm{Com}^{\scriptscriptstyle{W}}(\Xi)=1_{\mathcal{H}_{0}}\otimes B(\mathcal{K})$ hence

[TABLE]

where the last tensor product is the spatial tensor product of the $C^{*}$-algebras $K(\mathcal{H}_{0})$ and $B(\mathcal{K})$. Now the assertion (1) of the theorem follows from [28, Th. 3.8].

If $W$ is of finite multiplicity then $B(\mathcal{K})=K(\mathcal{K})$ hence $\mathrm{Com}(\Xi)=1_{\mathcal{H}_{0}}\otimes K(\mathcal{K})$ and thus $\mathscr{F}(\Xi)\cdot\mathrm{Com}(\Xi)=K(\mathcal{H}_{0})\otimes K(\mathcal{K})=K(\mathcal{H})$. Thus due to (1) an operator $T\in B(\mathcal{H})$ is compact if and only if $\lim_{\xi\to 0}\|(W(\xi)-1)T^{(*)}\|=0$. But we have to eliminate the condition on $T^{*}$ in order to get the assertion (b) of (3).

The implications $\Rightarrow$ in (a) and (b) of (3) are clear for any $W$ because $W$ is strongly continuous. Now we prove the implications $\Leftarrow$ assuming (a) is known in the irreducible case.

Assume $W$ of finite multiplicity and let $\{e_{1},\dots,e_{n}\}$ an orthonormal basis in the finite dimensional Hilbert space $\mathcal{K}$. Then any $h\in\mathcal{H}$ can be uniquely written as $h=\sum_{i}h_{i}\otimes e_{i}$ with $h_{i}\in\mathcal{H}_{0}$ and $\|h\|^{2}=\sum_{i}\|h_{i}\|^{2}$ hence $\|(W(\xi)-1))h\|^{2}=\sum_{i}\|(W_{0}(\xi)-1)h_{i}\|^{2}$ Let $\mathcal{J}\subset\mathcal{H}$ and $\mathcal{J}_{i}=\{h_{i}\mid h\in\mathcal{J}\}\subset\mathcal{H}_{0}$. If $\lim_{\xi\to 0}\sup_{h\in\mathcal{J}}\|(W(\xi)-1)h\|=0$ then we have $\lim_{\xi\to 0}\sup_{h_{i}\in\mathcal{J}_{i}}\|(W_{0}(\xi)-1)h_{i}\|=0$. Since part (a) of (3) is true in the case of irreducible representations, it follows that $\mathcal{J}_{i}$ is relatively compact in $\mathcal{H}_{0}$. Hence each $\mathcal{J}_{i}\otimes e_{i}$ is relatively compact in $\mathcal{H}$ so $\mathcal{J}\subset\sum_{i}\mathcal{J}_{i}\otimes e_{i}$ is relatively compact because a finite sum of relatively compact sets is relatively compact. This finishes the proof of part (a) of (3) for an arbitrary $W$ of finite multiplicity. Part (a) of (3) is an immediate consequence: if $T\in B(\mathcal{H})$ and $\lim_{\xi\to 0}\|(W(\xi)-1)T\|=0$ then $\lim_{\xi\to 0}\sup_{\|h\|\leq 1}\|(W(\xi)-1)Th\|=0$ hence the set $\{Th\mid\|h\|\leq 1\}$ is relatively compact which means that $T$ is compact.

Finally, assume $W$ is irreducible. If $\mathcal{J}$ is bounded then part (a) of (3) follows from [28, Theorem 3.4]. In order to eliminate the boundedness condition we use an improved Kolmogorov-Riesz criterion [47, 36]. We may assume that $\Xi$ is the phase space associated to a configuration space $X$ and that $W$ is the corresponding Schrödinger representation. Equip $X$ with an Euclidean structure and denote $\chi_{r}$ is the characteristic function of the region $|x|>r$. We prove first that the condition $\lim_{\xi\to 0}\sup_{u\in\mathcal{J}}\|W(\xi)h-h\|=0$ implies $\|\chi_{r}(q)h\|\to 0$ as $r\to\infty$ uniformly in $h\in\mathcal{J}$. This follows from the following more precise estimate which is of some independent interest: there is a number $c$ depending only on $\dim X$ such that for any $r>0$

[TABLE]

To prove it note that for any $\varepsilon>0$ we have

[TABLE]

In order to estimate the last integral for a fixed $x$ we identify $X=\mathbb{R}^{d}$ with the help of an orthonormal basis $e_{1},\dots,e_{d}$ such that $x=|x|e_{1}$. If we set $a_{1}=s$ and $(a_{2},\dots,a_{d})=t$ then the ball $|a|\leq\varepsilon$ is described by the conditions $-\varepsilon\leq s\leq\varepsilon$ and $t^{2}\leq\varepsilon^{2}-s^{2}$, hence there are numbers $C^{\prime},C^{\prime\prime}$ depending only on $d$ such that

[TABLE]

Then we get

[TABLE]

If $\varepsilon|x|>4$ then the last parenthesis is $>1/2$. Thus there is a number $C$ depending only on the dimension of $X$ such that for any $\varepsilon>0$ and any $h\in\mathcal{H}$

[TABLE]

This estimate is better than (3.10). Now the assertion (a) follows from [36, Theorem 1]. ∎

3.3. HVZ theorem

The original HVZ theorem, due to Hunziker, Van Winter and Zhislin, gives a description of the essential spectrum of a non-relativistic $N$-body Hamiltonian in terms of the spectra of its internal hamiltonians [1, 17, 22]. Proposition 8.4.2 from [1] is an abstract version of this theorem valid for $C^{*}$-algebras graded by finite semilattices that is very easy to prove and covers the usual versions of the theorem, see §9.4 and Ch. 10 of [1]. Our results here cover all the observables affiliated to the field algebra.

We begin with an abstract $C^{*}$-algebra versions of the HVZ theorem which follows from Theorems A.5 and A.8. Since $\Xi$ is finite dimensional $\mathscr{L}(\Xi)$ is an ideal of $\mathscr{L}$ and if $\mu\in\mathscr{L}$ its $\mathscr{L}$-essential spectrum $\mathscr{L}\textnormal{-}\mathrm{Sp_{ess}}(\mu)$ is defined by (A.6). In a finite multiplicity representation of $\Xi$ this is the essential spectrum of the operator corresponding to $\mu$.

Theorem 3.5**.**

The map $\mathscr{L}\to\bigoplus_{E\in\mathbb{H}(\Xi)}\mathscr{L}_{\scriptscriptstyle E}$ defined by \mu\mapsto\big{(}\mathcal{P}_{\scriptscriptstyle E}\mu\big{)}_{E\in\mathbb{H}(\Xi)} is a morphism with kernel $\mathscr{L}(\Xi)$ which gives an embedding $\mathscr{L}/\mathscr{L}(\Xi)\hookrightarrow\bigoplus_{E\in\mathbb{H}(\Xi)}\mathscr{L}_{\scriptscriptstyle E}$. For any $\mu\in\mathscr{L}$ the set $\{\mathcal{P}_{\scriptscriptstyle E}\mu\mid E\in\mathbb{H}(\Xi)\}$ is compact in $\mathscr{L}$ and

[TABLE]

Let $\mathcal{S}\subset\mathbb{G}(\Xi)$ a subsemilattice containing $\Xi$ and $\mathcal{S}_{\max}$ the set of maximal elements of $\mathcal{S}\setminus\{\Xi\}$. If $\mu\in\mathscr{L}(\mathcal{S})$ then

[TABLE]

If $\mathcal{S}$ is a countable set then $\mathcal{P}_{E}$ is expressed in terms of translations at infinity as in (2.30).

We give now a Hilbert space version of these algebraic results.

Lemma 3.6**.**

(1) If $\dim\Xi<\infty$ and $W$ is of finite multiplicity then $\mathscr{F}(\Xi)=\mathscr{F}\cap K(\mathcal{H})$.

(2) If $\dim\Xi<\infty$ and $W$ is of infinite multiplicity or $\dim\Xi=\infty$ then $\mathscr{F}\cap K(\mathcal{H})=0$.*

Proof.

If $W$ is a representation of an arbitrary symplectic space and if $\xi\neq 0$ the spectrum of $\phi(\xi)$ is equal to $\mathbb{R}$ and purely absolutely continuous, hence

[TABLE]

Thus if $T\in\mathscr{F}$ is compact then $\operatorname*{s-lim}_{r\to\infty}TW(r\xi)=0$ $\forall\xi\neq 0$ hence $\mathcal{P}_{\scriptscriptstyle E}T=0$ with the notations of Proposition 2.34. If $\dim\Xi<\infty$ we get $\mathcal{P}_{\scriptscriptstyle E}T=0$ for any hyperplane $E$ and then Theorem 3.5 implies $T\in\mathscr{F}(\Xi)$. Thus $\mathscr{F}\cap K(\mathcal{H})\subset\mathscr{F}(\Xi)$ and if $W$ of finite multiplicity then the inverse inclusion is true by Theorem 3.4. If $W$ is of infinite multiplicity then $\mathscr{F}\cap K(\mathcal{H})=0$ by Remark 2.31.

Now assume $\dim\Xi=\infty$ and let $T\in\mathscr{F}\cap K(\mathcal{H})$. Then $\mathcal{P}_{\scriptscriptstyle E}T=0$ if $E=\xi^{\sigma}$ with $\xi\neq 0$ by the argument at the beginning of this proof. If $\varepsilon>0$ there is $F\in\mathbb{G}(\Xi)$ and $S\in\mathscr{F}_{\scriptscriptstyle F}$ such that $\|S-T\|<\varepsilon$. If $\xi\in F^{\sigma}$ is nonzero then $F\subset\xi^{\sigma}=E$ hence $\mathcal{P}_{\scriptscriptstyle E}S=S$ and $\mathcal{P}_{\scriptscriptstyle E}T=0$ so $\|S\|<\varepsilon$ hence $\|T\|<2\varepsilon$. Since $\varepsilon$ is arbitrary we get $T=0$. ∎

Proposition 3.7**.**

If $\Xi$ is infinite dimensional or if $\Xi$ is finite dimensional and $W$ is of infinite multiplicity then $\mathscr{F}\cap K(\mathcal{H})=0$, hence $\mathrm{Sp_{ess}}(T)=\mathrm{Sp}(T)$ if $T\,\widetilde{\in}\,\mathscr{F}$.

This is a consequence of (A.8) and Lemma 3.6. Then we have a general HVZ type theorem:

Theorem 3.8**.**

Assume $\Xi$ finite dimensional and $W$ of finite multiplicity. If $T\,\widetilde{\in}\,\mathscr{F}$ then

[TABLE]

Moreover, for any $E\in\mathbb{H}(\Xi)$ and any nonzero $\xi\in E^{\sigma}$ we have

[TABLE]

Proof.

We will deduce this from Theorem 3.5 by taking into account the canonical isomorphisms between the algebras $\mathscr{L}$ and $\mathscr{F}$. For any operator $T\in\mathscr{F}$ let us denote $\Xi$-$\mathrm{Sp_{ess}}(T)$ the spectrum of the image of $T$ in the quotient algebra $\mathscr{F}/\mathscr{F}(\Xi)$. Then (3.12) give us

[TABLE]

Taking into account Proposition 2.34, it remains only to prove that \text{\small\Xi-}\mathrm{Sp_{ess}}(T) coincides with the essential spectrum of $T$ in the Hilbert space sense. But (1) of Lemma 3.6 implies that $\mathscr{F}/\mathscr{F}(\Xi)$ is the image of $\mathscr{F}$ in the Calkin algebra. ∎

The next result is an immediate consequence of Theorem 3.5 (by the same argument as above). In the simple case when $\mathcal{S}$ is finite this is a generalisation of the usual $N$-body HVZ theorem, cf. Ch. 10, §9.4 and Proposition 8.4.2 in [1].

Theorem 3.9**.**

Let $\Xi$ be finite dimensional and $W$ of finite multiplicity.If $\mathcal{S}\subset\mathbb{G}(\Xi)$ is a subsemilattice containing $\Xi$ and $\mathcal{S}_{\max}$ the set of maximal elements of $\mathcal{S}\setminus\{\Xi\}$, then

[TABLE]

One may use Theorem 3.8 to show that some operators are not affiliated to $\mathscr{F}$. Note first the following consequence of (3.16).

Corollary 3.10**.**

Assume $\Xi$ finite dimensional and $W$ of finite multiplicity. Then if $T\in\mathscr{F}$ we have $\operatorname*{s-lim}_{r\to\infty}W(r\xi)^{*}TW(r\xi)=0$ $\forall\xi\neq 0$ if and only if $T$ is a compact.

From this it follows that the generator of the dilation group associated to a Lagrangian decomposition of $\Xi$ is a self-adjoint operator not affiliated to $\mathscr{F}$. We prove this for $\dim\Xi=2$, the general case is similar. We may assume $\Xi=T^{*}\mathbb{R}$ and we work in the Schrödinger representation on $\mathcal{H}=L^{2}(\mathbb{R})$. Then $(qp+pq)/2$ is essentially self-adjoint on $C_{\mathrm{c}}^{\infty}(\mathbb{R})$ and its closure is a self-adjoint operator $\omega$ such that $(\mathrm{e}^{\mathrm{i}t\omega}f)(x)=\mathrm{e}^{t/2}f(\mathrm{e}^{t}x)$, so $\omega$ is the generator of the dilation group on $\mathcal{H}$. Clearly $\mathrm{e}^{\mathrm{i}t\omega}q\mathrm{e}^{-\mathrm{i}t\omega}=\mathrm{e}^{t}q$ and $\mathrm{e}^{\mathrm{i}t\omega}p\mathrm{e}^{-\mathrm{i}t\omega}=\mathrm{e}^{-t}p$ and the field operators are of the form $aq+bp$ with $a,b\in\mathbb{R}$ hence $\mathrm{e}^{\mathrm{i}t\omega}\mathscr{F}\mathrm{e}^{-\mathrm{i}t\omega}=\mathscr{F}$ $\forall t\in\mathbb{R}$ and the dilations induce a group of automorphisms of $\mathscr{F}$.

Proposition 3.11**.**

The self-adjoint operator $\omega$ is not affiliated to $\mathscr{F}$.

Proof.

We will prove that

[TABLE]

If $\omega$ is affiliated to $\mathscr{F}$ then $(\omega+\mathrm{i})^{-1}\in\mathscr{F}$. Then by (3.19) and Corollary 3.10 the operator $(\omega+\mathrm{i})^{-1}$ is compact which is impossible because the spectrum of $\omega$ is purely absolutely continuous. It remains to prove (3.19). Note that

[TABLE]

Indeed, since $\mathrm{i}(\omega+\mathrm{i})^{-1}=\int_{0}^{\infty}\mathrm{e}^{\mathrm{i}t\omega-t}\mathrm{d}t$ this follows by a change of variables in

[TABLE]

Now let $\xi=(x,k)$. Then

[TABLE]

hence

[TABLE]

Let us denote $\mathcal{I}_{r}$ the last integral over $y$. Then

[TABLE]

We have to show that $\langle{W(r\xi)g}|{\mathrm{i}(\omega+\mathrm{i})^{-1}W(r\xi)f}\rangle\to 0$ if $r\to\infty$ for any $f,g\in L^{2}(\mathbb{R})$. Clearly it suffices to assume $f,g$ continuous with compact support and then, by the preceding estimate and the Lebesgue dominated convergence theorem, it suffices to prove that $\mathcal{I}_{r}\to 0$ for any $\lambda>1$. Since $\xi\neq 0$, if $x=0$ then $k\neq 0$ then $\mathcal{I}_{r}\to 0$ by the Riemann-Lebesgue lemma. If $x\neq 0$ and $r$ is large enough then $\bar{g}(y)f(\lambda y+(1-\lambda)rx)=0\ \forall y$ hence $\mathcal{I}_{r}=0$. This proves the assertion. ∎

4. Subalgebras of $\mathscr{F}$ and

Hamiltonians affiliated to them

4.1. Half-Lagrangian decompositions and magnetic

fields

The decompositions of a symplectic space considered here and the representations associated to them will enable us to describe rather explicitly certain subalgebras of interest of $\mathscr{F}$.

Let $X$ be a finite dimensional real vector space, $X^{*}$ its dual, and $\langle{\cdot},{\cdot}\rangle:X\times X^{*}\to\mathbb{R}$ the duality form. Then the Fourier transform333 This definition and that of the scalar product in $L^{2}(X)$ require the choice of a Lebesgue measure $\mathrm{d}x$ on $X$ but the choice is irrelevant in the construction of the observables and algebras of interest. $(Fu)(k)=\int_{X}\mathrm{e}^{-\mathrm{i}\langle{x},{k}\rangle}u(x)\mathrm{d}x$ induces an isomorphism of $L^{2}(X)$ onto $L^{2}(X^{*})$. The position and momentum observables $q$ and $p$ are defined as follows: if $\varphi$ and $\psi$ are (equivalences classes of) complex Borel functions on $X$ and $X^{*}$ then $\varphi(q)=M_{\varphi}$ and $\psi(p)=F^{-1}M_{\psi}F$ where $M_{\varphi}$ and $M_{\psi}$ are the operators of multiplication by $\varphi$ in $L^{2}(X)$ and by $\psi$ in $L^{2}(X^{*})$. We set

[TABLE]

and identify functions $\varphi\in L^{\infty}(X),\psi\in L^{\infty}(X^{*})$ with operators $\varphi(q),\psi(p)\in\mathbf{B}(X)$, so

[TABLE]

If $X$ is an Euclidean space $X^{*}$ is identified with $X$ as usual but the spaces $L^{\infty}(X)$ and $L^{\infty}(X^{*})$ are different when viewed as subspaces of $\mathbf{B}(X)$: the first one is the space of $\varphi(q)$ while the second is the space of $\varphi(p)$ with $\varphi\in L^{\infty}(X)$. Or $L^{\infty}(X^{*})=F^{-1}L^{\infty}(X)F$.

If $k\in X^{*}$ and $x\in X$ let $\langle{q},{k}\rangle=\varphi(q)$ with $\varphi=\langle{\cdot},{k}\rangle$ and $\langle{x},{p}\rangle=\psi(p)$ with $\psi=\langle{x},{\cdot}\rangle$. These are self-adjoint operators on $\mathcal{H}$ such that $[k(q),x(p)]=\mathrm{i}\langle{x},{k}\rangle$ and

[TABLE]

The phase space of $X$ is the cotangent space $T^{*}X=X\oplus X^{*}$ equipped with the symplectic form $\sigma(\xi,\eta)=\langle{y},{k}\rangle-\langle{x},{l}\rangle$ if $\xi=x+k,\eta=y+l$ with $x,y\in X$ and $k,l\in X^{*}$, cf. [39, §21.1]. A simple modification of this symplectic form allows us to introduce constant magnetic fields into the formalism and so to treat N-body systems which interact with an external asymptotically constant magnetic field (we refer to [35] for a detailed study of such systems). The constant magnetic field may be interpreted as a bilinear anti-symmetric form $\beta:X\times X\to\mathbb{R}$ and then clearly the next relation defines a symplectic form on $T^{*}X$:

[TABLE]

In physical terms $X$ is the configuration space of a system, $T^{*}X$ is its phase space, and $\beta$ represents an external constant magnetic field. A $C^{*}$-algebraic approach to these questions, including non constant magnetic fields, was proposed in [29].

We reconsider this construction in the symplectic setting. A finite dimensional symplectic space $\Xi$ is even dimensional, say $\dim\Xi=2n$, and if $E,F$ are subspaces of $\Xi$:

(1) an isotropic $E$ is Lagrangian if and only if $\dim E=n$;

(2) $\{E\mid E\text{ is Lagrangian}\}$ is a smooth submanifold of $\mathbb{G}(\Xi)$ of dimension $n(n+1)/2$;

(3) $E$ is isotropic (coisotropic) $\Leftrightarrow$ $E$ is contained in (contains) a Lagrangian subspace;

(4) $\dim E=n$ $\Rightarrow$ there is a Lagrangian $F$ such that $\Xi=E\oplus F$;

(5) $E$ is symplectic $\Leftrightarrow E^{\sigma}$ is symplectic $\Leftrightarrow\Xi=E+E^{\sigma}\Leftrightarrow\Xi=E\oplus E^{\sigma}$.

Definition 4.1**.**

A half-Lagrangian decomposition of $\Xi$ is a couple $(X,X^{*})$ of subspaces of $\Xi$ such that $\dim X=\dim\Xi/2$, $X^{*}$ is Lagrangian, and $X\cap X^{*}={0}$. If $X$ is also Lagrangian we call $(X,X^{*})$ a Lagrangian decomposition.

Under the conditions of the definition we have $\Xi=X\oplus X^{*}$ and we usually say that $\Xi=X\oplus X^{*}$ is a half-Lagrangian or Lagrangian decomposition of $\Xi$. Let $\beta$ be the bilinear anti-symmetric form on $X$ defined by the restriction $\beta=\sigma|X\times X$. If $\xi,\eta\in\Xi$ and $\xi=x+k,\eta=y+l$ are their decompositions in $X\oplus X^{*}$ and since $X^{*}$ is isotropic

[TABLE]

A half-Lagrangian decomposition is Lagrangian if and only if $\beta=0$.

Since $X^{*}$ is isotropic the map $k\ni X^{*}\mapsto\sigma(k,\cdot)$ is an injective, hence bijective, map of $X^{*}$ onto the dual of $X$ and we will identify these spaces by setting

[TABLE]

Then (4.5) becomes (4.4). So the choice of a half-Lagrangian decomposition amounts to choosing the configuration space $X$ of a system in an external constant magnetic field $\beta$. The decomposition is Lagrangian if and only if the magnetic field is zero.

If $W$ is any representation of $\Xi$ on a Hilbert space $\mathcal{H}$, for $x\in X,k\in X^{*}$ and we set

[TABLE]

Then if $x,y\in X$ and $k,l\in X^{*}$ the CCR (1.11) imply

[TABLE]

The Schrödinger representation associated to the preceding half-Lagrangian decomposition is the irreducible representation of $\Xi$ acting in $\mathcal{H}=L^{2}(X)$ as follows:

[TABLE]

Clearly this may be rewritten as

[TABLE]

From $[\varphi(q),\langle{x},{p}\rangle]=\mathrm{i}(x\varphi^{\prime})(q)$ if $\varphi$ is $C^{1}$ and $x\varphi^{\prime}$ its derivative in the direction $x$, we get

[TABLE]

in particular $[\langle{x},{p}\rangle,\beta(x,q)]=\mathrm{i}\beta(x,x)=0$. Thus, in terms of (4.7), we have

[TABLE]

Moreover, the Baker-Campbell-Hausdorff formula gives

[TABLE]

Here $\phi(\xi)$ is the field operator at the point $\xi=x+k\in\Xi$.

We keep the notation $\beta$ for the map $X\ni z\mapsto\beta(\cdot,z)\in X^{*}$; this is the unique linear map $\beta:X\to X^{*}$ such that $\beta(x,y)=\langle{x},{\beta y}\rangle$ for all $x,y\in X$. Then $\beta^{*}:X^{**}=X\to X^{*}$ and the anti-symmetry of the bilinear form $\beta$ is equivalent to $\beta^{*}=-\beta$. Clearly $\beta(x,q)=\langle{x},{\beta q}\rangle$ where $\beta q$ is the map $X\to X^{*}$ defined by $(\beta q)(y)=\beta y=\beta(\cdot,y)$. Thus, if we set $p_{\beta}=p+\frac{1}{2}\beta q$, then the field operator may be written $\phi(\xi)=\langle{q},{k}\rangle+\langle{x},{p_{\beta}}\rangle$.

We will now prove that the Hamiltonian of a non-relativistic system of particles in a constant magnetic field is affiliated to $\mathscr{F}$. With the help of the perturbative criterion of Theorem A.11 one may then show that Hamiltonians of $N$-body systems with singular $n$-body interactions ($1\leq n\leq N$) and asymptotically constant external magnetic fields are affiliated to $\mathscr{F}$. The arguments are similar to those of §4.3 but we do not develop this topic (this could be done along the lines of [29, §4]). The next lemma is valid in any irreducible representation of a finite dimensional symplectic space $\Xi$.

Lemma 4.2**.**

Let $\eta=\{\eta_{1},\dots,\eta_{n}\}$ be a generating set for the subspace $E\in\mathbb{G}(\Xi)$. Then the form sum $\Delta_{\eta}=\sum_{k=1}^{n}\phi(\eta_{k})^{2}$ is a self-adjoint operator affiliated to $\mathscr{F}(E)$.

Proof.

We show that $T=(\Delta_{\eta}+1)^{-1}\in\mathscr{F}(E)$ by checking the three conditions of Theorem 1.11. Note that the form domain of $\Delta_{\eta}$ is $\mathcal{K}=\textstyle{\bigcap}_{k}\mathrm{dom}(\phi(\mathrm{e}_{k}))$. We embed as usual $\mathcal{K}\subset\mathcal{H}=\mathcal{H}^{*}\subset\mathcal{K}^{*}$. By (1.11) the operators $W(\xi)$ leave invariant $\mathcal{K}$ so extend to bounded operators in $\mathcal{K}^{*}$. Below we consider $T$ as an operator $\mathcal{H}\rightarrow\mathcal{K}$ and $\mathcal{K}^{*}\rightarrow\mathcal{H}$ (by taking adjoints) and $\Delta_{\eta}$ as operator $\mathcal{K}\rightarrow\mathcal{K}^{*}$. Clearly $[W(\xi),T]=T[\Delta_{\eta},W(\xi)]T$ and

[TABLE]

If $\xi\in E^{\sigma}$ the last term is zero, so the last condition of Theorem 1.11 is satisfied. Then

[TABLE]

hence the first condition of Theorem 1.11 is satisfied too. Now observe that if $\xi=\sum_{k=1}^{n}\xi_{k}$ an induction argument gives

[TABLE]

If $\xi\in E$ then we can write it as $\xi=\sum t_{k}\eta_{k}$ with real $t_{k}$. By using the preceding formula we see that in order to prove the second condition of Theorem 1.11 it suffices to show that $\|[W(t\eta_{k})-1]T\|\rightarrow 0$ when $t\rightarrow 0$ for each $k$. But

[TABLE]

so the assertion follows immediately from the estimate $\phi(\eta_{k})^{2}\leq\Delta_{\eta}$. ∎

The case of interest here is $E=X$; recall that $\Xi=X\oplus X^{*}$ is a half-Lagrangian decomposition. If $\eta$ is a basis of $X$ then $\Delta_{\eta}$ is the “free” Hamiltonian of a system with configuration space $X$ in the constant magnetic field $\beta=\sigma|X\times X$. More general such “free” Hamiltonians are the self-adjoint operators affiliated to the $C^{*}$-algebra $\mathscr{F}(X)$ which, by (4.7) and (4.13), is the norm closure in $\mathbf{B}(X)$ of the set of operators

[TABLE]

We could write this as $\hat{u}(p_{\beta})$ and think of the elements of $\mathscr{F}(X)$ as functions of class $C_{0}$ of $p_{\beta}$, but this is rather formal because the components of the vector-operator $p_{\beta}$ do not commute: if $\beta\neq 0$ then $\{W_{x}^{p}\}_{x\in X}$ is only a projective representation of $X$, cf. (4.8).

Example 4.3**.**

The simplest physically interesting example is a non-relativistic particle with configuration space $X=\mathbb{R}^{3}$ interacting with the constant magnetic field $b\in X$. If $x,y\in X$ let $x\times y$ be their cross product and define the bilinear anti-symmetric form $\beta:X\times X\to\mathbb{R}$ by $\beta(x,y)=\langle{x\times y},{b}\rangle$. The symplectic space will be the phase space $\Xi=X\oplus X$ equipped with the symplectic form (4.4), i.e.

[TABLE]

Since $\beta(x,y)=\langle{x\times y},{b}\rangle=\langle{x},{y\times b}\rangle$ we get $\beta q=q\times b$ hence $p_{\beta}=p+\frac{1}{2}q\times b$. From (4.14) we get for $\xi=(e,0)\equiv e\in X$

[TABLE]

If $e_{1},e_{2},e_{3}$ the natural basis of $X=\mathbb{R}^{3}$ then $\langle{e_{j}},{q\times b}\rangle=(q\times b)_{j}$ and

[TABLE]

hence $\phi(e_{j})=p_{j}+\textstyle{\frac{1}{2}}(q_{k}b_{l}-q_{l}b_{k})$ with usual notation rules. Then

[TABLE]

Recall that the vector field $a:X\to X$ defined by $a(x)=\frac{1}{2}b\times x=-\frac{1}{2}x\times b$ is the magnetic vector potential, i.e. $\nabla\times a=b$. Thus $p_{\beta}=p-a$ and

[TABLE]

which is the Hamiltonian of a non-relativistic particle with mass $2$ and charge $1$ interacting with the constant magnetic field $b\in X$. Also $H=p_{\beta}^{2}$ with the usual rule that the square of $p_{\beta}$ is the sum of the squares of its components ($p_{\beta}$ is the magnetic momentum).

Remark 4.4**.**

In the preceding example we worked with the cotangent space $T^{*}X=X\oplus X$ but modified its symplectic form, cf. (4.15). Since the field algebras associated to symplectic spaces of dimension $6$ are isomorphic, we should be able to work in $T^{*}X$ with the standard symplectic form (1.36) (i.e. (4.4) with $\beta$ replaced by [math]) and this is indeed possible. Now the field operator at the point $\xi=(x,k)$ is $\phi(x,k)=\langle{q},{k}\rangle+\langle{x},{p}\rangle$. We take $\eta_{j}=(\textstyle{\frac{1}{2}}b\times e_{j},e_{j})\in T^{*}X$ with $j=1,2,3$ and denote $E$ the subspace they generate. Clearly $\phi(\eta_{j})=p_{j}+\textstyle{\frac{1}{2}}(q_{k}b_{l}-q_{l}b_{k})$ which are the same operators as in Example 4.3 hence $H=\sum_{j=1}^{3}\phi(\eta_{j})^{2}=(p-a)^{2}$ as before but this time $H$ is affiliated to $\mathscr{F}(E)$.

4.2. Subalgebras of

$\mathscr{F}$

The following examples of $C^{*}$-subalgebras of the field algebra play a role in a generalized version of the N-body problem. By Remark 2.31 it suffices to work in an irreducible representation of $\Xi$. The symplectic space is finite dimensional and the subalgebras that we construct depend on a Lagrangian decomposition $\Xi=X\oplus X^{*}$ of $\Xi$, cf. Definition 4.1. By assertions (2) and (4) in §4.1 there are infinitely many such choices and the algebras constructed below depend on the choices.

We fix a Lagrangian decomposition $\Xi=T^{*}X$ and we work in the Schrödinger representation associated to it, see §4.1 for framework and notations of. Thus $\mathcal{H}=L^{2}(X)$, the field operator at the point $\xi=(x,k)$ is $\phi(\xi)=\langle{x},{p}\rangle+\langle{q},{k}\rangle$, and the $C^{*}$-algebra generated by these operators is the field algebra $\mathscr{F}\subset\mathbf{B}(X)=B(L^{2}(X))$.

Recall that by (4.2) we identify $C_{\mathrm{b}}^{\mathrm{u}}(X)$ and $C_{\mathrm{b}}^{\mathrm{u}}(X^{*})$ with the $C^{*}$-subalgebras of $\mathbf{B}(X)$ consisting of the operators $\varphi(q)$ and $\psi(p)$ respectively, with $\varphi\in C_{\mathrm{b}}^{\mathrm{u}}(X)$ and $\psi\in C_{\mathrm{b}}^{\mathrm{u}}(X^{*})$.

If $Y\in\mathbb{G}(X)$ then $C_{0}(X/Y)\subset C_{\mathrm{b}}^{\mathrm{u}}(X)$ as in §1.2. If $Z\in\mathbb{G}(X^{*})$ then we also have $C_{0}(Z^{*})\subset C_{\mathrm{b}}^{\mathrm{u}}(X)$ because $Z^{*}=X/Z^{\perp}$ with $Z^{\perp}=\{x\in X\mid\langle{x},{k}\rangle=0\ \forall k\in Z\}$. Thus

[TABLE]

On the other hand, if $Y^{\perp}=\{k\in X^{*}\mid\langle{x},{k}\rangle=0\ \forall x\in Y\}$ then $Y^{*}=X^{*}/Y^{\perp}$. Hence

[TABLE]

The Grassmann $C^{*}$-algebras $\mathcal{G}_{\scriptscriptstyle{X}}$ and $\mathcal{G}_{\scriptscriptstyle X^{*}}$ defined in (1.8) are embedded in $\mathbf{B}(X)$ as above.

We may now give “explicit” descriptions of some components of $\mathscr{F}$.

Proposition 4.5**.**

If $Y\in\mathbb{G}(X)$ and $Z\in\mathbb{G}(X^{*})$ then

[TABLE]

In particular $\mathscr{F}(X)=C_{0}(X^{*})$, $\mathscr{F}(X^{*})=C_{0}(X)$, and for any $Y\subset X$

[TABLE]

Moreover

[TABLE]

Proof.

$\mathscr{F}(Y)$ is the norm closure of the set of operators $W(\mu)=\int_{Y}W(y)u(y)\mathrm{d}y$ with $u\in L^{1}(Y)$, cf. Definition 2.27. Then (4.11) gives $W(\mu)=\int_{Y}\mathrm{e}^{\frac{\mathrm{i}}{2}\langle{y},{p}\rangle}u(y)\mathrm{d}y=c\hat{u}(p)$ where $c$ is a constant depending on the definition of the Fourier transform. Since $\hat{u}\in C_{0}(Y^{*})\subset C_{\mathrm{b}}^{\mathrm{u}}(X^{*})$ we get $\mathscr{F}(Y)=C_{0}(Y^{*})$. By the same definition and with the notation (1.3) we have $\mathscr{F}(Y)=C^{*}(\langle{y_{1}},{p}\rangle)\cdot C^{*}(\langle{y_{2}},{p}\rangle)\cdot\ldots\cdot C^{*}(\langle{y_{n}},{p}\rangle)$ where $y_{1},\dots,y_{n}$ is a basis of $Y$, which gives a second proof of the same result. If $Z\subset X^{*}$ the computation of $\mathscr{F}(Z)$ is similar: if we take $\xi=(x,k)=(0,z)$ in (4.11) we get $W(\xi)=\mathrm{e}^{\mathrm{i}\langle{q},{z}\rangle}$ hence if $u\in L^{1}(Y)$ then $W(\mu)=\int_{Y}\mathrm{e}^{\mathrm{i}\langle{q},{z}\rangle}u(y)\mathrm{d}y=c\hat{u}(q)$ where $\hat{u}\in C_{0}(Z^{*})$ and $Z^{*}=X/Z^{\perp}$, cf. the comments before the statement of the proposition. Then by using (1.18) and we get

[TABLE]

The first relation in (4.19) follows from the second one in (4.18) by taking $Z=Y^{\perp}$ since $(Y^{\perp})^{\perp}=Y$. The second relation in (4.19) is a consequence of the third one in (4.18) since $Y^{\sigma}=X+Y^{\perp}$ but it is instructive to give a direct proof. $\mathscr{F}(Y^{\sigma})$ is the closure of the set of operators $W(\mu)=\int_{Y^{\sigma}}W(\xi)\mu(\mathrm{d}\xi)$ with $\mu\in L^{1}(Y^{\sigma})$. If $\mu(\xi)=\mathrm{e}^{\frac{\mathrm{i}}{2}\langle{x},{k}\rangle}u(x)v(k)$ with $u\in C_{\mathrm{c}}(X)$ and $v\in C_{\mathrm{c}}(Y^{\perp})$ then by (4.11)

[TABLE]

where $\hat{u}\in C_{0}(X^{*})$ and $\hat{v}\in C_{0}(X/Y)$ are the Fourier transforms of $u$ and $v$, we identify $Y^{\perp}=(X/Y)^{*}$, and $c$ is a constant depending on the normalization chosen in the definition of the Fourier transforms. Thus we have

[TABLE]

This implies $\mathscr{F}(Y^{\sigma})=C_{0}(X/Y)\cdot C_{0}(X^{*})$ because the linear space generated by the functions $\mu$ is dense in $L^{1}(Y^{\sigma})$. Finally, $Z\mapsto Z^{\perp}$ being a bijective map $\mathbb{G}(X^{*})\to\mathbb{G}(X)$, (4.18) implies (4.20) and (4.21). ∎

Remark 4.6**.**

In terms of the Grassmannian $C^{*}$-algebras introduced in §1.2, we have

[TABLE]

where $\mathcal{G}_{\scriptscriptstyle{X}}\rtimes X$ is the crossed product of $\mathcal{G}_{\scriptscriptstyle{X}}$ by the action of $X$, cf. [31]. On the other hand, (1.24) implies $\mathscr{F}_{{\scriptscriptstyle{X}}}=C^{*}(\phi(\xi)\mid\xi\in X)$ and $\xi\in X$ means $\xi=(x,0)$ with $x\in X$ so $\phi(\xi)=\langle{x},{p}\rangle+\langle{q},{k}\rangle=\langle{x},{p}\rangle$. With a similar argument in the case of $X^{*}$, we get

[TABLE]

This and (4.23) give the first assertion of Proposition 1.1.

Proposition 4.7**.**

$L^{\infty}(X)\cap\mathscr{F}=\mathscr{F}_{\scriptscriptstyle X^{*}}=\mathcal{G}_{\scriptscriptstyle{X}}$* and $L^{\infty}(X^{*})\cap\mathscr{F}=\mathscr{F}_{{\scriptscriptstyle{X}}}=\mathcal{G}_{\scriptscriptstyle X^{*}}$.*

Proof.

Note that we use the embeddings (4.2). The inclusion $L^{\infty}(X)\cap\mathscr{F}\supset\mathcal{G}_{\scriptscriptstyle{X}}$ is obvious. And $L^{\infty}(X)\cap\mathscr{F}\subset\mathscr{F}_{\scriptscriptstyle X^{*}}^{\mathrm{com}}=\mathscr{F}_{\scriptscriptstyle X^{*}}=\mathcal{G}_{\scriptscriptstyle{X}}$ by (4.21) and Corollary 2.37. ∎

The algebras $\mathscr{F}(Y^{\sigma})$ with $Y\in\mathbb{G}(X)$ are components of the algebras generated by N-body type Hamiltonians, cf. §4.3, hence it is useful to have “explicit” descriptions of them. Below we give two such descriptions, consequences of Theorem 1.11 and Proposition 4.5.

Proposition 4.8**.**

If $Y\subset X$ and $T\in\mathbf{B}(X)$ then $T\in\mathscr{F}(Y^{\sigma})$ if and only if

1. (i)

$[\mathrm{e}^{\mathrm{i}\langle{x},{p}\rangle},T]=0$* for all $x\in Y$,* 2. (ii)

${\displaystyle\lim_{x\to 0}}\|[\mathrm{e}^{\mathrm{i}\langle{x},{p}\rangle},T]\|=0$* and ${\displaystyle\lim_{x\to 0}}\|(\mathrm{e}^{\mathrm{i}\langle{x},{p}\rangle}-1)T\|=0$,* 3. (iii)

${\displaystyle\lim_{k\to 0}}\|[\mathrm{e}^{\mathrm{i}\langle{q},{k}\rangle},T]\|=0$* and ${\displaystyle\lim_{k\to 0,k\in Y^{\perp}}}\|(\mathrm{e}^{\mathrm{i}\langle{q},{k}\rangle}-1)T\|=0$.*

Remark 4.9**.**

The second condition in (ii) is equivalent to $T=\varphi(p)T_{0}$ for some $\varphi\in C_{0}(X^{*})$ and $T_{0}\in\mathbf{B}(X)$, cf. [32, Lemma 3.8].

In the next proposition $F_{Y}$ is the Fourier transform associated to $Y$ and we keep the notation $F_{Y}$ for $F_{Y}\otimes 1:L^{2}(Y)\otimes L^{2}(Y^{\prime})\to L^{2}(Y^{*})\otimes L^{2}(Y^{\prime})\equiv L^{2}(Y^{*};L^{2}(Y^{\prime}))$.

Proposition 4.10**.**

Let $Y,Y^{\prime}$ be subspaces of $X$ such that $X=Y\oplus Y^{\prime}$ and let us identify $L^{2}(X)=L^{2}(Y)\otimes L^{2}(Y^{\prime})$. Then

[TABLE]

Thus $T\in\mathbf{B}(X)$ belongs to $\mathscr{F}(Y^{\sigma})$ if and only if $F_{Y}TF_{Y}^{-1}$ is the operator of multiplication by a function $\widehat{T\,}\in C_{0}(Y^{*};\mathbf{K}(Y^{\prime}))$ in $L^{2}(Y^{*};L^{2}(Y^{\prime}))$.

Proof.

We identify $X/Y=Y^{\prime}$ hence $C_{0}(X^{*})=C_{0}(Y^{*}\oplus Y^{\prime*})=C_{0}(Y^{*})\otimes C_{0}(Y^{\prime*})$ and $C_{0}(Y^{\prime})\cdot C_{0}(Y^{\prime*})=\mathbf{K}(Y^{\prime})$. Then by (4.19)

[TABLE]

4.3. N-body Hamiltonians and beyond

Hamiltonians generalizing N-body Hamiltonians are defined by their affiliation to graded $C^{*}$-subalgebras $\mathscr{F}(\mathcal{S})$ where $\mathcal{S}\subset\mathbb{G}(\Xi)$ is a subsemilattice. The case of finite $\mathcal{S}$ is easy but important because large classes of N-body type Hamiltonians are affiliated to such algebras. By (3) of Theorem 2.9 and (2) of Lemma A.1, if $\mathcal{S}$ is a finite subset of $\mathbb{G}(\Xi)$ then $\mathscr{F}(\mathcal{S})$ is a subalgebra of $\mathscr{F}$ if and only if $\mathcal{S}$ is a subsemilattice and then $\mathscr{F}(\mathcal{S})$ is an $\mathcal{S}$-graded $C^{*}$-subalgebra of $\mathscr{F}$. However, we are mostly interested in the case of infinite $\mathcal{S}$.

In this section we fix a Lagrangian decomposition $\Xi=X\oplus X^{*}$ and consider semilattices of the form $\mathcal{S}=\mathcal{T}^{\sigma}$ with $\mathcal{T}\subset\mathbb{G}(X)$ stable under intersections. We recall the notations

[TABLE]

Since $Y\mapsto Y^{\perp}$ and $Y\mapsto Y^{\sigma}$ are bijective maps $\mathbb{G}(X)\to\mathbb{G}(X^{*})$ and $\mathbb{G}(X)\to\mathbb{G}_{\scriptscriptstyle\supset X}(\Xi)$ which reverse the order, we get

[TABLE]

Recall that $\mathcal{G}_{\scriptscriptstyle{X}}$ is the Grassmann C*-algebra, cf. §1.2. Taking into account (4.19), we have

[TABLE]

These are stable under translations and conjugation subspace of $C_{\mathrm{b}}^{\mathrm{u}}(X)$ and if $\mathcal{T}$ is finite then $\mathring{\mathcal{G}}_{\scriptscriptstyle{X}}(\mathcal{T})$ is closed, so $\mathring{\mathcal{G}}_{\scriptscriptstyle{X}}(\mathcal{T})=\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T})$. And $\mathcal{G}_{\scriptscriptstyle{X}}(\{Y\})\equiv\mathcal{G}_{\scriptscriptstyle{X}}(Y)=C_{0}(X/Y)$.

Our first goal is to show that $\mathscr{F}(\mathcal{S})$ is the $C^{*}$-algebra generated by Hamiltonians of N-body type. Clearly:

Lemma 4.11**.**

Let $\mathcal{T}_{0}\subset\mathbb{G}(X)$ and let $\mathcal{T}$ be the set of intersections of subspaces from $\mathcal{T}_{0}$. Then the unital $C^{*}$-subalgebra of $C_{\mathrm{b}}^{\mathrm{u}}(X)$ generated by $\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T}_{0})$ is $\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T})$.

Remark 4.12**.**

An intersection of subspaces of $\mathcal{T}_{0}$ is a subspace of the form $\cap_{i\in I}Y_{i}$ with $Y_{i}\in\mathcal{T}_{0}$. Since $X$ is finite dimensional we may assume $I$ finite, but it could be empty and then $\cap_{i\in\emptyset}Y_{i}=X$, hence we always have $X\in\mathcal{T}$. Thus $\mathbb{C}=C_{0}(X/X)\subset\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T})$.**

We refer to [30, 31, 32] for a presentation of the notion of crossed product of a $C^{*}$-algebra by the action of a group adapted to our context (see also Proposition 2.38). If $\mathcal{A}\subset C_{\mathrm{b}}^{\mathrm{u}}(X)$ is a $C^{*}$-subalgebra stable under translations then the norm closed linear space $\mathcal{A}\cdot C_{0}(X^{*})$ of operators on $L^{2}(X)$ generated by the products $\varphi(q)\psi(p)$ with $\varphi\in\mathcal{A}$ and $\psi\in C_{0}(X^{*})$ is a $C^{*}$-algebra canonically isomorphic to the crossed product $\mathcal{A}\rtimes X$, cf. [31, Th. 2.17]. We identify $\mathcal{A}\rtimes X=\mathcal{A}\cdot C_{0}(X^{*})$ so one may take this as definition of the crossed product.

A function $h:X^{*}\to\mathbb{R}$ is called divergent if $\lim_{k\to\infty}h(k)=+\infty$.

Theorem 4.13**.**

Let $\mathcal{T}_{0}\subset\mathbb{G}(X)$ and $\mathcal{T}$ the set of intersections of subspaces from $\mathcal{T}_{0}$. Then $\mathcal{S}=\mathcal{T}^{\sigma}$ is a subsemilattice of $\mathbb{G}_{\scriptscriptstyle\supset X}(\Xi)$ and

[TABLE]

If $h:X^{*}\to\mathbb{R}$ is continuous and divergent $\mathscr{F}(\mathcal{S})$ coincides with the $C^{*}$-algebra generated by the self-adjoint operators $H=h(p+k)+v(q)$ with $k\in X^{*}$ and $v\in\mathring{\mathcal{G}}_{\scriptscriptstyle{X}}(\mathcal{T}_{0})$ real.

Proof.

By (1.22) and (4.19) we have

[TABLE]

which proves (4.26). By Remark 4.12 we have $C_{0}(X^{*})\subset\mathscr{F}(\mathcal{S})$ hence for an arbitrary $h:X^{*}\to\mathbb{R}$ continuous and divergent the self-adjoint operator $h(p)$ is affiliated to $C_{0}(X^{*})$ hence to $\mathscr{F}(\mathcal{S})$. Then if $v\in\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T})$ real $H=h(p)+v(q)$ is a self-adjoint operator and for large enough negative $z$ we have a norm convergent development

[TABLE]

But $(z-h(p))^{-1}\in C_{0}(X^{*})$ hence $v(q)(z-h(p))^{-1}\in\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T})\cdot C_{0}(X^{*})$ and $(z-H)^{-1}$ belongs to $\mathscr{F}(\mathcal{S})$. Thus any $h(p)+v(q)$ is affiliated to $\mathscr{F}(\mathcal{S})$ and it remains to prove that for any fixed $h$ the algebra $\mathscr{F}(\mathcal{S})$ is equal to the algebra $\mathscr{C}$ generated by operators of the form $H=h(p+k)+v(q)$ with $k\in X^{*}$ and $v\in\mathring{\mathcal{G}}_{\scriptscriptstyle{X}}(\mathcal{T}_{0})$ real. Clearly we get the same $\mathscr{C}$ if we allow $v\in\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T}_{0})$ and the preceding argument gives $\mathscr{C}\subset\mathscr{F}(\mathcal{S})$.

We will follow now the proof of [30, Pr. 4.1] with $\mathcal{A}=\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T}_{0})$. Then $\mathcal{A}$ is not necessarily a $C^{*}$-subalgebra but is a closed stable under translations and conjugation subspace of $C_{\mathrm{b}}^{\mathrm{u}}(X)$. Then the quoted proof gives $v(q)\psi(p)\in\mathscr{C}$ for all $v\in\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T}_{0})$ and $\psi\in C_{0}(X^{*})$. In other terms, $\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T}_{0})\cdot C_{0}(X^{*})\subset\mathscr{C}$ which in fact is equivalent to $\mathcal{G}_{\scriptscriptstyle{X}}(Y)\cdot C_{0}(X^{*})\subset\mathscr{C}$ for all $Y\in\mathcal{T}_{0}$. Observe that $\mathcal{G}_{\scriptscriptstyle{X}}(Y)\cdot C_{0}(X^{*})=C_{0}(X^{*})\cdot\mathcal{G}_{\scriptscriptstyle{X}}(Y)$ because this is the $C^{*}$-algebra $\mathscr{F}(Y^{\perp})$. Then for any $Y,Z\in\mathcal{T}_{0}$

[TABLE]

hence $\mathcal{G}_{\scriptscriptstyle{X}}(Y\cap Z)\cdot C_{0}(X^{*})\subset\mathscr{C}$. Thus $\mathcal{G}_{\scriptscriptstyle{X}}(Y)\cdot C_{0}(X^{*})\subset\mathscr{C}$ for all $Y\in\mathcal{T}$ which is clearly equivalent to $\mathscr{F}(\mathcal{S})=\mathcal{G}_{\scriptscriptstyle{X}}(\mathcal{T})\cdot C_{0}(X^{*})\subset\mathscr{C}$. ∎

Corollary 4.14**.**

If $h:X^{*}\to\mathbb{R}$ is continuous and divergent then

[TABLE]

Remark 4.15**.**

If $\mathcal{S}\subset\mathbb{G}_{\scriptscriptstyle\supset X}(\Xi)$ is a subsemilattice then $\mathscr{F}(\mathcal{S})\cdot C_{\mathrm{b}}(X^{*})=\mathscr{F}(\mathcal{S})$ by (4.26). Thus if $T\in\mathscr{F}(\mathcal{S})$ and $\psi\in C_{\mathrm{b}}(X^{*})$ then $\psi(p)T$ and $T\psi(p)$ belong to $\mathscr{F}(\mathcal{S})$.

We will apply this proposition in the case of a system of $N$ particles moving in $\mathbb{R}^{\nu}$. Then $X=\mathbb{R}^{N\nu}$, we write $x=(x_{1},\dots,x_{N})$ with $x_{i}\in\mathbb{R}^{\nu}$, and denote $q_{i},p_{i}$ the momentum and position observables of the $i$-th particle, hence $q=(q_{1},\dots,q_{N})$, $p=(p_{1},\dots,p_{N})$ (if $\nu>1$ this is not the notation from §1.7). If the particles interact only via one-body and two-body forces the Hamiltonian is

[TABLE]

where $h_{1},\dots,h_{N}$ are real continuous divergent functions on $\mathbb{R}^{\nu}$ and $h_{i}(p_{i}-k_{i})$ is interpreted as the kinetic energy of the i-th particle if we take $k_{i}$ as origin in the momentum space. The potentials $v_{i},v_{ij}$ are real functions in $C_{0}(\mathbb{R}^{\nu})$ but we may write them in the form required in Proposition 1.26 as follows. Define $\rho_{i},\rho_{ij}:X\to\mathbb{R}^{\nu}$ by $\rho_{i}(x)=x_{i}$ and $\rho_{ij}(x)=x_{j}-x_{i}$ for $i<j$. These are linear surjective maps with $\ker\rho_{i}=X_{i}=\{x\mid x_{i}=0\}$ and $\ker\rho_{ij}=X_{ij}=\{x\mid x_{i}=x_{j}\}$, so $\rho_{i}$ and $\rho_{ij}$ induce bijective linear maps $X/X_{i}\to\mathbb{R}^{\nu}$ and $X/X_{ij}\to\mathbb{R}^{\nu}$. Then $v_{i}(q_{i})=v_{i}\circ\rho_{i}(q)$ and $v_{ij}(q_{i}-q_{j})=v_{ij}\circ\rho_{ij}(q)$ hence by keeping the notation $v_{i}$ for $v_{i}\circ\rho_{i}$ and $v_{ij}$ for $v_{ij}\circ\rho_{ij}$ the potential part in (4.27) becomes $\sum v_{i}(q)+\sum v_{ij}(q)$ where $v_{i}\in C_{0}(X/X_{i})$ and $v_{ij}\in C_{0}(X/X_{ij})$. Then the $C^{*}$-algebra generated by the Hamiltonians (4.27) is given by Proposition 1.26: $\mathcal{T}_{0}$ is the set of subspaces $X_{i},X_{ij}$ hence it suffices to compute the set of intersections of such subspaces. This is an exercice, we describe the result below.

Let $[N]=\{1,2,\dots,N\}$. A nonempty subset of $[N]$ is called cluster. A sub-partition of $[N]$ is a (possibly empty) set of pairwise disjoint clusters; if $a$ is a sub-partition then $\cup a$ is the union of its clusters. We denote $\pi$ the set of sub-partitions. If $a\in\pi$ and $i,j\in[N]$ we write $i\stackrel{{\scriptstyle a}}{{\sim}}j$ if $i,j$ belong to the same cluster of $a$ and we define

[TABLE]

Then $\mathcal{T}=\{X_{a}\mid a\in\pi\}$. Thus, *if we set $\mathcal{S}=\{X_{a}^{\sigma}\mid a\in\pi\}$ then the $C^{*}$-algebra generated by the Hamiltonians (4.27) with arbitrary $k_{i}\in\mathbb{R}^{\nu}$ and $v_{i},v_{ij}\in C_{0}(\mathbb{R}^{\nu})$ is *

[TABLE]

Note that, although we started with Hamiltonians involving only one and two body interactions, the Hamiltonians affiliated to $\mathscr{F}(\mathcal{S})$ may involve $k$-body interactions with any $k\leq N$, and much more in fact. We discus this question below.

We fix a subsemilattice $\mathcal{S}\subset\mathbb{G}(\Xi)$ with $\min\mathcal{S}=X$ and a continuous positive and divergent function $h:X^{*}\to\mathbb{R}$. Then $h(p)$ is a kinetic energy operator affiliated to $\mathscr{F}(X)=C_{0}(X^{*})$ and our goal is to build self-adjoint operators $H=h(p)+V$ affiliated to $\mathscr{F}(\mathcal{S})$. We use the terminology and results of §A.4. Let $\mathcal{H}_{h}=D(h(p)^{1/2})$ be the form domain of $h(p)$ and $\mathcal{H}_{h}\subset\mathcal{H}\subset\mathcal{H}_{h}^{*}$ the associated scale. Theorem A.11 and Corollary 1.17 imply:

Theorem 4.16**.**

*For each $E\in\mathcal{S}$ let $V(E)\in B(\mathcal{H}_{h},\mathcal{H}_{h}^{*})$ a symmetric operator such that

(1) $V(X)=0$ and $\varphi(p)V(E)(h(p)+1)^{-1/2}\in\mathscr{F}(E)$ $\forall\varphi\in C_{\mathrm{c}}(X^{*})$,

(2) the family $\{V(E)\}_{E\in\mathcal{S}}$ is norm summable in $B(\mathcal{H}_{h},\mathcal{H}_{h}^{*})$,

(3) $V(E)\geq-\mu_{\scriptscriptstyle E}h(p)-\nu_{\scriptscriptstyle E}$ with $\mu_{\scriptscriptstyle E},\nu_{\scriptscriptstyle E}\geq 0$, $\sum_{E\in\mathcal{S}}\mu_{\scriptscriptstyle E}<1$, and $\sum_{E\in\mathcal{S}}\nu_{\scriptscriptstyle E}<\infty$.

Let $V=\sum_{E\in\mathcal{S}}V(E)$ and $V_{\scriptscriptstyle E}=\sum_{F\leq E}V(F)$ for any $E\in\mathcal{S}$. Then the form sums $H=h(p)+V$ and $H_{\scriptscriptstyle E}=h(p)+V_{\scriptscriptstyle E}$ are bounded from below self-adjoint operators, with form domain $\mathcal{H}_{h}$, strictly affiliated to $\mathscr{F}(\mathcal{S})$, and $\mathcal{P}_{\scriptscriptstyle E}H=H_{\scriptscriptstyle E}\ \forall E\in\mathcal{S}$. If $\Xi\in\mathcal{S}$ and $\mathcal{S}_{\max}$ is the set of maximal elements of $\mathcal{S}\setminus\{\Xi\}$ then*

[TABLE]

We give explicit versions of condition (1) of Theorem 4.16 under stronger assumptions on $h$. Since $E=Y^{\sigma}$ with $Y\in\mathbb{G}(X)$ we have $T=\varphi(p)V(E)(h(p)+1)^{-1/2}\in\mathscr{F}(E)$ if and only if $T$ satisfies the conditions of one of the Propositions 4.8 or 4.10. Since $\mathrm{e}^{\mathrm{i}\langle{x},{p}\rangle}$ and $(h(p)+1)^{-1/2}$ are commuting operators, $\{\mathrm{e}^{\mathrm{i}\langle{x},{p}\rangle}\}_{x\in X}$ induces strongly continuous unitary groups in $\mathcal{H}_{h}$ and $\mathcal{H}_{h}^{*}$, but $\mathcal{H}_{h}$ is not stable under the operators $\mathrm{e}^{\mathrm{i}\langle{q},{k}\rangle}$. In order to simplify later statements we will impose a rather strong growth condition on $h$ which ensures $\mathrm{e}^{\mathrm{i}\langle{q},{k}\rangle}\mathcal{H}_{h}\subset\mathcal{H}_{h}$; the condition can be relaxed so as to cover hypoelliptic operators for example (see [20, §4.3]) but this is not really relevant in our context.

We equip $X$ with an Euclidean structure, denote $|\cdot|$ its norm, set $\langle{x}\rangle=(1+|x|^{2})^{1/2}$, and identify $X^{*}=X$. Then $\mathcal{H}^{s}\equiv\mathcal{H}^{s}(X)$ is the Sobolev space of order $s\in\mathbb{R}$ defined by the norms $\|u\|_{s}=\|\langle{p}\rangle^{s}u\|$. Then $\mathcal{H}^{0}=\mathcal{H}$, $\mathcal{H}^{s}\subset\mathcal{H}^{t}$ if $s\geq t$, and $\mathcal{H}^{s}\subset\mathcal{H}\subset\mathcal{H}^{-s}=(\mathcal{H}^{s})^{*}$ if $s\geq 0$. Clearly $\{\mathrm{e}^{\mathrm{i}\langle{x},{p}\rangle}\}_{x\in X}$ and $\{\mathrm{e}^{\mathrm{i}\langle{q},{k}\rangle}\}_{k\in X}$ are strongly continuous $C_{0}$-groups of bounded operators in $\mathcal{H}^{s}$ for any real $s$.

Definition 4.17**.**

If $s\geq 0$ let $\mathbf{B}^{s}(X)\doteq B(\mathcal{H}^{s},\mathcal{H}^{-s})$ and

[TABLE]

If $T\in\mathbf{B}_{0}^{s}(X)$ we say that $T:\mathcal{H}^{s}\to\mathcal{H}^{-s}$ is small at infinity.

If the condition in (4.31) is satisfied for one $t$ it is satisfied for any $t$. The elements of $\mathbf{B}_{0}^{s}(X)$ are the operators $T\in\mathbf{B}^{s}(X)$ which decay at infinity in a weak sense. For example, $\varphi(q)\in\mathbf{B}^{0}(X)$ if and only if $\varphi:X\to\mathbb{C}$ is a bounded Borel function and $\varphi(q)\in\mathbf{B}_{0}^{0}(X)$ means that this function satisfies $\lim_{a\to\infty}\int_{|x-a|}|\varphi(x)|\mathrm{d}x=0$ [21].

If $h$ is a real functions on $X$ and $s>0$ is a number we write $h(x)\sim|x|^{2s}$ if

[TABLE]

From now on we assume that this condition is satisfied. Clearly this is equivalent to $\mathcal{H}_{h}=\mathcal{H}^{s}$. Thus $\mathcal{H}_{h}^{*}=\mathcal{H}^{-s}$ and the $V(E)$ from Theorem 4.16 are symmetric operators $V(E)\in\mathbf{B}^{s}(X)$. Recall that $E=Y^{\sigma}$ for some $Y\in\mathbb{G}(X)$.

Lemma 4.18**.**

Let $V(E)\in\mathbf{B}^{s}(X)$ such that $[\mathrm{e}^{\mathrm{i}\langle{x},{p}\rangle},V(E)]=0\ \forall x\in Y$. Assume that there is $t>s$ such that the following conditions hold in norm in $B(\mathcal{H}^{s},\mathcal{H}^{-t})$:

[TABLE]

Then $\varphi(p)V(E)(h(p)+1)^{-1/2}\in\mathscr{F}(Y^{\sigma})$ for any $\varphi\in C_{\mathrm{c}}(X)$.

Proof.

We have

[TABLE]

hence by (4.33) and Remark 4.15 (with $\mathcal{S}=\{E\}$) we have for any $\varphi\in C_{\mathrm{c}}(X)$

[TABLE]

$T=\varphi(p)V(E)\langle{p}\rangle^{-s}\in\mathscr{F}(Y^{\sigma})$ if and only if $T$ satisfies the conditions of Proposition 4.8 and it is easy to check that they are equivalent to those in the statement of the lemma. ∎

Let $Y^{\prime}$ be a complementary subspace of $Y$ in $X$. Then $X=Y\oplus Y^{\prime}$ and we identify

[TABLE]

Let $F_{\scriptscriptstyle{Y}}$ be the Fourier transform associated to $Y$ acting in $L^{2}(Y)$ and keep the notation $F_{\scriptscriptstyle{Y}}$ for $F_{\scriptscriptstyle{Y}}\otimes 1$ which acts in $L^{2}(X)=L^{2}(Y)\otimes L^{2}(Y^{\prime})$. Let $q_{\scriptscriptstyle{Y}},p_{\scriptscriptstyle{Y}}$ be the position and momentum observables associated to $Y$, so $\varphi(p_{\scriptscriptstyle{Y}})=F_{\scriptscriptstyle{Y}}^{-1}\varphi(q_{\scriptscriptstyle{Y}})F_{\scriptscriptstyle{Y}}$ for any Borel function $\varphi:Y\to\mathbb{C}$, hence $C_{0}(Y^{*})=F_{\scriptscriptstyle{Y}}^{-1}C_{0}(Y)F_{\scriptscriptstyle{Y}}$ (see §1.7 and §4.1). Then by (4.25)

[TABLE]

Thus if $\Phi:Y\to\mathbf{K}(Y^{\prime})$ is of class $C_{0}$ and we denote $\Phi(q_{\scriptscriptstyle{Y}})$ the operator of multiplication by $\Phi$ in $L^{2}(X)=L^{2}(Y;L^{2}(Y^{\prime}))$, then $\Phi(p_{\scriptscriptstyle{Y}})\doteq F_{\scriptscriptstyle{Y}}^{-1}\Phi(q_{\scriptscriptstyle{Y}})F_{\scriptscriptstyle{Y}}\in\mathscr{F}(Y^{\sigma})$. This definition extends to functions on $Y$ with values operators on Sobolev spaces on $Y^{\prime}$.

Lemma 4.19**.**

Let $V_{\scriptscriptstyle{Y}}:Y\to\mathbf{B}_{0}^{s}(Y^{\prime})$ norm continuos and such that for any $y$ the operator $V_{\scriptscriptstyle{Y}}(y):\mathcal{H}^{s}(Y^{\prime})\to\mathcal{H}^{-s}(Y^{\prime})$ is symmetric and satisfies $\pm V_{\scriptscriptstyle{Y}}(y)\leq c_{Y}\langle{|y|+|p_{Y^{\prime}}|}\rangle^{2s}$ for some constant $c_{Y}$. Then $\varphi(p)V_{\scriptscriptstyle{Y}}(p_{\scriptscriptstyle{Y}})(h(p)+1)^{-1/2}\in\mathscr{F}(Y^{\sigma})$ for any $\varphi\in C_{\mathrm{c}}(X)$.

Indeed, a straightforward computation shows that $V(E)=V_{\scriptscriptstyle{Y}}(p_{\scriptscriptstyle{Y}})$ satisfies the conditions of Lemma 4.18. Alternatively, Lemma 4.19 is a consequence of Proposition 4.10, cf. the proof of [21, Th. 4.6]. Theorem 1.30 is a consequence Theorem 4.16 and Lemma 4.19.

Theorem 4.20**.**

Let $\mathcal{S}\subset\mathbb{G}(\Xi)$ a subsemilattice with $\min\mathcal{S}=X$ and $h:X\to\mathbb{R}$ continuous with $h(x)\sim|x|^{2s}$ for some $s>0$. Let $V:\mathcal{H}^{s}\to\mathcal{H}^{-s}$ symmetric such that $V\geq-\mu h(p)-\nu$ with $\mu<1,\nu\geq 0$ and satisfying the following equivalent conditions:

[TABLE]

Then the form sum $H=h(p)+V$ is a self-adjoint operator strictly affiliated to $\mathscr{F}(\mathcal{S})$.

Proof.

We first prove the equivalence in (4.37). Since $\varphi(p)\in\mathscr{F}(\mathcal{S})$ the implication $\Rightarrow$ is clear. Assume that the right hand side is satisfied and let $\varphi$ be a continuous function on $X$ such that $0\leq\varphi\leq 1$, $\varphi(x)=1$ if $|x|<1$, and $\varphi(x)=0$ if $|x|>2$. If $\varepsilon>0$ then $\varphi(\varepsilon p)V\langle{p}\rangle^{-s}\in\mathscr{F}(\mathcal{S})$ and

[TABLE]

The second term on the left hand side belongs to $\mathscr{F}(\mathcal{S})$ and

[TABLE]

The right hand side above clearly tends to zero when $\varepsilon\to 0$ hence $\langle{p}\rangle^{-t}V\langle{p}\rangle^{-s}\in\mathscr{F}(\mathcal{S})$.

We will use the notations and results of Theorem A.9. The self-adjoint operator $H_{0}=h(p)$ is bounded from below and strictly affiliated to $\mathscr{F}(X)=C_{0}(X^{*})$ hence strictly affiliated to $\mathscr{F}(\mathcal{S})$. Its form domain is $\mathcal{H}^{s}$ whose adjoint space is $\mathcal{H}^{-s}$ hence $V$ is a form perturbation of $H_{0}$, hence the form sum $H=H_{0}+V$ is a bounded from below self-adjoint operator. If $\varphi\in C_{\mathrm{c}}(\mathbb{R})$ then $\psi=\varphi\circ h\in C_{\mathrm{c}}(X)$ and $\varphi(H_{0})=\psi(p)$ so $\varphi(H_{0})V(|H_{0}|+1)^{-1/2}\in\mathscr{F}(\mathcal{S})$. Indeed, we have

[TABLE]

and $\langle{p}\rangle^{s}(|h(p)|+1)^{-1/2}=\theta(p)$ for some $\theta\in C_{\mathrm{b}}(X)$ so the right hand side above belongs to $\mathscr{F}(\mathcal{S})\cdot C_{\mathrm{b}}(X)=\mathscr{F}(\mathcal{S})$ by Remark 4.15. So the result follows from Theorem A.9. ∎

5. Extended field C*-algebra

Let $\Xi$ be a symplectic space and $W$ a representation of $\Xi$ on a Hilbert space $\mathcal{H}$. The field $C^{*}$-algebra $\mathscr{F}$ as defined in (1) of Definition 2.27 contains, in some sense, only functions of the fields $\phi(\xi)$. If the representation $W$ is not irreducible then many other physically interesting observables are not affiliated to $\mathscr{F}$, e.g. the Hamiltonians involving spin interactions. Here we introduce a $\mathbb{G}(\Xi)$-graded $C^{*}$-algebra $\mathscr{E}$ of operators on $\mathcal{H}$ which contains $\mathscr{F}$ and fixes this flaw.

In the next definition, suggested by Theorem 1.11, we introduce the components of $\mathscr{E}$.

Definition 5.1**.**

*If $E\in\mathbb{G}(\Xi)$ then $\mathscr{E}(E)\equiv\mathscr{E}_{\scriptscriptstyle\Xi}^{\scriptscriptstyle{W}}(E)$ is the set of $T\in B(\mathcal{H})$ such that:

(i)   $\|[W(\xi),T]\|\to 0$ if $\xi\to 0$ in $\Xi$,

(ii)  $[W(\xi),T]=0$ if $\xi\in E^{\sigma}$,

(iii) $\|(W(\xi)-1)T\|\to 0$ if $\xi\in E$ and $\xi\rightarrow 0$.

In particular $\mathscr{E}(0)=\mathrm{Com}(\Xi)$.*

If $\dim\Xi=\infty$ the limit in (i) is defined in the following equivalent ways:

(1) for any $\xi\in\Xi$ we have $\lim_{r\to 0}\|[W(r\xi),T]\|=0$,

(2) for any $F\in\mathbb{G}(\Xi)$ we have $\lim_{\xi\in F,\xi\to 0}\|[W(\xi),T]\|=0$.

Thus (i) can be reformulated as: the map $\xi\mapsto W(\xi)TW(\xi)^{*}$ is norm continuous on finite dimensional subspaces of $\Xi$. To prove (1)$\Rightarrow$(2) note that $\|[W(\xi),T]\|=\|\mathscr{W}(\xi)T-T\|$ where $\mathscr{W}(\xi)$ is the automorphism of $B(\mathcal{H})$ defined by $\mathscr{W}(\xi)T=W(\xi)TW(\xi)^{*}$ and if $e_{1},\dots,e_{n}$ is a basis in $F$ and $\xi=\sum_{i}r_{i}e_{i}$ then $\mathscr{W}(\xi)=\mathscr{W}(r_{1}e_{1})\dots\mathscr{W}(r_{n}e_{n})$.

Proposition 5.2**.**

Assume $\Xi$ finite dimensional. Then $\mathscr{E}(\Xi)=\mathscr{F}(\Xi)\cdot\mathrm{Com}(\Xi)$ and $W$ is of finite multiplicity if and only if $\mathscr{E}(\Xi)=K(\mathcal{H})$.

Proof.

If $\Xi$ is finite dimensional and we take $E=\Xi$ then only (i) and (iii) are nontrivial and (i)+(iii) is equivalent to $\lim_{\xi\to 0}\|(W(\xi)-1)T^{(*)}\|=0$ hence by (1) of Theorem 3.4 we have $\mathscr{E}(\Xi)=\mathscr{F}(\Xi)\cdot\mathrm{Com}(\Xi)$. Then we use (3) of Theorem 3.4. ∎

Proposition 5.3**.**

$\mathscr{E}(E)$* is a non-degenerate $C^{*}$-subalgebra of $B(\mathcal{H})$ and $\mathscr{F}(E)\subset\mathscr{E}(E)$.*

Proof.

It is easily seen that $\mathscr{E}(E)$ is a norm closed subalgebra of $B(\mathcal{H})$. If $T\in\mathscr{E}(E)$ then clearly $\|[W(\xi),T^{*}]\|\to 0$ if $\xi\to 0$ in $\Xi$ and $[W(\xi),T^{*}]=0$ if $\xi\in E^{\sigma}$. Moreover if $\xi\rightarrow 0$ in $E$ then $\|T^{*}(W(\xi)-1)\|\to 0$ hence also $\|(W(\xi)-1)T^{*}\|\to 0$ because $\|[W(\xi),T^{*}]\|\to 0$. Thus $T^{*}\in\mathscr{E}(E)$ hence $\mathscr{E}(E)$ is a $C^{*}$-algebra. Below we show $\mathscr{F}(E)\subset\mathscr{E}(E)$ and since $\mathscr{F}(E)$ is non-degenerate, $\mathscr{E}(E)$ is also non-degenerate.

To prove the last assertion of the proposition it suffices to show that $W(\mu)\in{\mathscr{E}}(E)$ if $\mu\in L^{1}(E)$. For any $\xi\in\Xi$ and $\mu\in M(E)$ we have

[TABLE]

Thus $\|[W(\xi),W(\mu)]\|\rightarrow 0$ if $\xi\rightarrow 0$. Moreover, the above commutator is clearly zero if $\xi\in E^{\sigma}$. It remains to show that $W(\mu)$ verifies (iii) of Definition 5.1 if $\mu\in L^{1}(E)$. Then $\mu(\mathrm{d}\eta)=\rho(\eta)\mathrm{d}_{E}\eta$ with $\rho$ is a function on $E$. If $\xi\in E$ we get from (1.11):

[TABLE]

Clearly both integrals tend to zero as $\xi\rightarrow 0$ in $E$. ∎

Remark 5.4**.**

If $W$ is of finite multiplicity then $\mathscr{E}(E)=\mathscr{F}(E)\cdot\mathrm{Com}(\Xi)$: the inclusion $\supset$ is obvious and the proof of the converse is similar to that of Theorem 1.11 given in §3.1.**

Now let $\rho$ be an integrable function on $E$ such that $\int_{E}\rho(\eta)\mathrm{d}_{E}\eta=1$ for some Lebesgue measure $\mathrm{d}_{E}\eta$ on $E$. For $\varepsilon>0$ set $\rho_{\varepsilon}(\eta)=\varepsilon^{-m}\rho(\eta/\varepsilon)$, where $m=\dim E$, and

[TABLE]

Lemma 5.5**.**

(i)* $W(\rho_{\varepsilon})\in{\mathscr{E}}(E)$ and $\operatorname*{s-lim}_{\varepsilon\rightarrow 0}W(\rho_{\varepsilon})=1$ on $\mathcal{H}$.

                 (ii) If $T\in{\mathscr{E}}(E)$ then $\lim_{\varepsilon\rightarrow 0}\|(W(\rho_{\varepsilon})-1)T\|=0$.*

Proof.

For (i), see the proof of Proposition 2.32-(a). Then, by Definition 5.1-(iii) we have

[TABLE]

Lemma 5.6**.**

If $\xi\notin E^{\sigma}$, and $T\in\mathscr{E}(E)$ then $\operatorname*{s-lim}_{r\to\infty}TW(r\xi)=0$

Proof.

By Lemma 5.5 we have $\lim_{\varepsilon\to 0}TW(\rho_{\varepsilon})=T$ in norm hence it suffices to have $\operatorname*{s-lim}_{r\to\infty}SW(r\xi)=0$ for $S\in\mathscr{F}(E)$ which is true by Lemma 2.33. ∎

Theorem 5.7**.**

The family of $C^{*}$-subalgebras $\{\mathscr{E}(E)\}_{E\in\mathbb{G}(\Xi)}$ is linearly independent and satisfies $\mathscr{E}(E)\cdot\mathscr{E}(F)\subset\mathscr{E}(E+F)$.

Proof.

The proof of the linear independence of the family of $\mathscr{E}(E)$ is similar to the proof of Theorem 2.9-(1). Let $\mathcal{S}$ be a finite set of finite dimensional subspaces of $\Xi$ and for each $F\in\mathcal{S}$ let $T(F)\in\mathscr{E}(F)$ such that $\sum_{F\in\mathcal{S}}T(F)=0$; we have to show $T(E)=0$ $\forall E$. If $F\in\mathcal{S}$ and $F\not\subset E$ then $E^{\sigma}\not\subset F^{\sigma}$ hence $E^{\sigma}\cap F^{\sigma}$ is a strict subspace of $E^{\sigma}$. Then we may choose $\xi\in E^{\sigma}$ which does not belong to any of these subspaces, i.e. $\xi\notin F^{\sigma}$ if $F\in\mathcal{S}$ is not a subset of $E$, which implies $\operatorname*{s-lim}_{r\to\infty}T(F)W(r\xi)=0$ for all such $F$ by Lemma 5.6. On the other hand, if $F\subset E$ then $\xi\in E^{\sigma}\subset F^{\sigma}$ hence $W(r\xi)^{*}T(F)W(r\xi)=T(F)$ by (ii) of Definition 5.1. By taking $r\to\infty$ in the identity $\sum_{F\in\mathcal{S}}W(r\xi)^{*}T(F)W(r\xi)=0$ we thus get $\sum_{F\subset E}T(F)=0$ for all $E\in\mathcal{S}$. If $E$ is minimal in $\mathcal{S}$ this implies $T(E)=0$. Then $\sum_{F\in\mathcal{S}_{1}}\mu(F)=0$ if $\mathcal{S}_{1}$ is the set of elements of $\mathcal{S}$ which are not minimal. By repeating the above argument for $\mathcal{S}_{1}$ we get $T(E)=0$ for all $E$ minimal in $\mathcal{S}_{1}$, etc.

Now we prove $\mathscr{E}(E)\cdot\mathscr{E}(F)\subset\mathscr{E}(E+F)$. Let $S\in\mathscr{E}(E)$ and $T\in\mathscr{E}$, we must show $ST\in\mathscr{E}(E+F)$. Since $(E+F)^{\sigma}=E^{\sigma}\cap F^{\sigma}$, the conditions (i) and (ii) of Definition 5.1 are obviously satisfied, it remains to show $\|(W(\xi)-1)ST\|\to 0$ if $\xi\to 0$ in $E+F$. If $E\not\subset F$ let $G$ be subspace of $F$ supplementary to $E\cap F$, hence $E+F=E+G$ and $E\cap G=0$. Then any $\xi\in\ E+F$ has a unique decomposition $\xi=\eta+\zeta$ with $\eta\in E,\zeta\in G$ and $\xi\to 0$ in $E+F$ is equivalent to $\eta\to 0$ in $E$ and $\zeta\to 0$ in $G$ (hence in $F$). Then

[TABLE]

and the first term on the right hand side tends in norm to zero as $\zeta\to 0$ while $\mathrm{e}^{\frac{\mathrm{i}}{2}\sigma(\zeta,\eta)}\to 1$ and $W(\eta)SW(\zeta)T\to ST$ in norm if $\eta\to 0$ and $\zeta\to 0$. ∎

Definition 5.8**.**

The extended field $C^{*}$-algebra associated to the representation $W$ is

[TABLE]

and is a $\mathbb{G}(\Xi)$-graded $C^{*}$-algebra of bounded operators on $\mathcal{H}$.

In Proposition 5.2 we already gave an example where the algebra $\mathscr{E}$ is more natural than $\mathscr{F}$. We give one more example involving coupling of finite multiplicity representations, which covers for example N-body Dirac and Pauli operators.

Let $\Xi_{1},\dots,\Xi_{N}$ finite dimensional symplectic spaces equipped with finite multiplicity representations $W_{1},\dots,W_{N}$ on Hilbert space $\mathcal{H}_{1},\dots,\mathcal{H}_{N}$. The coupling of these systems is defined as follows: $\Xi\doteq\Xi_{1}\oplus\dots\oplus\Xi_{N}$ (symplectic direct sum) equipped with the natural representation on $\mathcal{H}\doteq\mathcal{H}_{1}\otimes\dots\otimes\mathcal{H}_{N}$ defined by

[TABLE]

Then the extended field algebra $\mathscr{E}$ associated to $(\Xi,W)$ contains $\mathscr{E}_{1}\otimes\dots\otimes\mathscr{E}_{N}$ and Hamiltonians affiliated to it may be constructed with the help of Theorem A.9 starting with “free” Hamiltonians of the form $H_{1}\otimes 1\otimes\dots\otimes 1+\dots+1\otimes\dots\otimes 1\otimes H_{N}$.

Appendix A C*-algebras graded by Grassmannians

A.1. Main facts

Let $\Xi$ be a real vector space and $\mathscr{C}$ a $C^{*}$-algebra graded by a subsemilattice $\mathcal{S}$ of $\mathbb{G}(\Xi)$. We use the definitions and notations of §1.1, thus $\mathscr{C}=\sum_{E\in\mathcal{S}}^{\mathrm{c}}\mathscr{C}(E)$ for a given family $\{\mathscr{C}(E)\}_{E\in\mathcal{S}}$ of $C^{*}$-subalgebras of $\mathscr{C}$ satisfying $\mathscr{C}(E)\mathscr{C}(F)\subset\mathscr{C}(E+F)$ for all $E,F\in\mathcal{S}$ and such that the linear sum $\mathring{\mathscr{C}}\doteq\sum_{E\in\mathcal{S}}\mathscr{C}(E)$ is direct and dense in $\mathscr{C}$.

If $\mathcal{T}\subset\mathcal{S}$ we set $\mathring{\mathscr{C}}(\mathcal{T})=\sum_{E\in\mathcal{T}}\mathscr{C}(E)$ and $\mathscr{C}(\mathcal{T})=\sum_{E\in\mathcal{T}}^{\mathrm{c}}\mathscr{C}(E)$. If $\mathcal{T}$ is a subsemilattice then clearly $\mathscr{C}(\mathcal{T})$ is a $\mathcal{T}$-graded $C^{*}$-algebra with components $\mathscr{C}(E)$; subalgebras of this form are called graded $C^{*}$-subalgebras of $\mathscr{C}$. Below we mention some properties of $\mathscr{C}(\mathcal{T})$: (1) is [43, Pr. 1.6], (2) is an exercise, and (3) is [21, Pr. 3.2].

Lemma A.1**.**

(1)* If $\mathcal{T}$ is finite then $\sum_{E\in\mathcal{T}}\mathscr{C}(E)$ is closed, hence $\mathscr{C}(\mathcal{T})=\sum_{E\in\mathcal{T}}\mathscr{C}(E)$.

(2) If $\mathcal{T}$ is finite and $\mathscr{C}(E)\cdot\mathscr{C}(F)=\mathscr{C}(E+F)$ for all $E,F\in\mathcal{T}$, then $\mathscr{C}(\mathcal{T})$ is a subalgebra of $\mathscr{C}$ if and only if $\mathcal{T}$ is a subsemilattice of $\mathcal{S}$.

(3) $\mathscr{C}=\bigcup_{\mathcal{T}}\mathscr{C}(\mathcal{T})$ union over all countable subsemilattices of $\mathcal{S}$.*

For any subspace $E$ of $\Xi$ the sets

[TABLE]

are subsemilattices. If $E\in\mathcal{S}$ the next theorem is [19, Th. 3.1] while [43, Pr. 1.10] is a more general result; the last assertion is clear if $T\in\mathring{\mathscr{C}}$ which is dense in $\mathscr{C}$, etc.

Theorem A.2**.**

$\mathscr{C}_{\scriptscriptstyle E}\doteq\mathscr{C}(\mathcal{S}_{\scriptscriptstyle E})$* is a $C^{*}$-subalgebra and $\mathscr{C}^{\prime}_{\scriptscriptstyle E}\doteq\mathscr{C}(\mathcal{S}^{\prime}_{\scriptscriptstyle E})$ an ideal of $\mathscr{C}$ such that $\mathscr{C}=\mathscr{C}_{\scriptscriptstyle E}+\mathscr{C}^{\prime}_{\scriptscriptstyle E}$ and $\mathscr{C}_{\scriptscriptstyle E}\cap\mathscr{C}^{\prime}_{\scriptscriptstyle E}=0$. The projection $\mathcal{P}_{\scriptscriptstyle E}:\mathscr{C}\to\mathscr{C}_{\scriptscriptstyle E}$ determined by this direct sum decomposition is a morphism. $\{\mathcal{P}_{\scriptscriptstyle E}T\mid E\in\mathcal{S}\}$ is relatively compact $\forall\ T\in\mathscr{C}$.*

The projection morphism (cf. §1.1) $\mathcal{P}_{\scriptscriptstyle E}$ is the unique continuous map $\mathscr{C}\to\mathscr{C}$ such that

[TABLE]

where $T(G)\in\mathscr{C}(G)$ and $T(G)=0$ but for a finite number of $G$. Clearly if $E\subset F$ then $\mathscr{C}_{\scriptscriptstyle E}\subset\mathscr{C}_{\scriptscriptstyle F}$ and $\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle F}=\mathcal{P}_{\scriptscriptstyle F}\mathcal{P}_{\scriptscriptstyle E}=\mathcal{P}_{\scriptscriptstyle E}$. More generally:

Proposition A.3**.**

If $E,F$ are subspaces of $\Xi$ then

[TABLE]

Proof.

We first prove the relation $\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle F}=\mathcal{P}_{\scriptscriptstyle E\cap F}$. Since $\mathscr{C}$ is the closure of $\sum_{G\in\mathcal{S}}\mathscr{C}(G)$ it suffices to show that the restrictions of the maps $\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle F}$ and $\mathcal{P}_{\scriptscriptstyle E\cap F}$ to each $\mathscr{C}(G)$ are equal. There are three possibilities:

(1) $G\subset E\cap F$: then each of the operators $\mathcal{P}_{\scriptscriptstyle E},\mathcal{P}_{\scriptscriptstyle F},\mathcal{P}_{\scriptscriptstyle E\cap F}$ is the identity on $\mathscr{C}(G)$ so $\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle F}=\mathcal{P}_{\scriptscriptstyle E\cap F}$ on $\mathscr{C}(G)$;

(2) $G\not\subset F$: then $\mathcal{P}_{\scriptscriptstyle F}\mathscr{C}(G)=\mathcal{P}_{\scriptscriptstyle E\cap F}\mathscr{C}(G)=0$ hence $\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle F}=\mathcal{P}_{\scriptscriptstyle E\cap F}$ on $\mathscr{C}(G)$;

(3) $G\subset F$ but $G\not\subset E$: then $\mathcal{P}_{\scriptscriptstyle F}$ is the identity on $\mathscr{C}(G)$, $\mathcal{P}_{\scriptscriptstyle E}\mathscr{C}(G)=0$, $\mathcal{P}_{\scriptscriptstyle E\cap F}\mathscr{C}(G)=0$ hence $\mathcal{P}_{\scriptscriptstyle E\cap F}=\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle F}$ on $\mathscr{C}(G)$.

Thus $\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle F}=\mathcal{P}_{\scriptscriptstyle E\cap F}$. The operator $\mathcal{P}_{\scriptscriptstyle E\cap F}$ is a projection of $\mathscr{C}$ onto $\mathscr{C}_{\scriptscriptstyle E\cap F}$ and we clearly have $\mathscr{C}_{\scriptscriptstyle E\cap F}\subset\mathscr{C}_{\scriptscriptstyle E}\cap\mathscr{C}_{\scriptscriptstyle F}$. On the other hand each of the operators $\mathcal{P}_{\scriptscriptstyle E},\mathcal{P}_{\scriptscriptstyle F}$ is the identity on $\mathscr{C}_{\scriptscriptstyle E}\cap\mathscr{C}_{\scriptscriptstyle F}$ hence $\mathcal{P}_{\scriptscriptstyle E\cap F}=\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle F}$ is also the identity on $\mathscr{C}_{\scriptscriptstyle E}\cap\mathscr{C}_{\scriptscriptstyle F}$ so $\mathscr{C}_{\scriptscriptstyle E\cap F}\supset\mathscr{C}_{\scriptscriptstyle E}\cap\mathscr{C}_{\scriptscriptstyle F}$. ∎

A.2. Quotient algebra

We now assume that the subsemilattice $\mathcal{S}$ has a greatest element $\sup\mathcal{S}$ or, equivalently, $\sup_{E\in\mathcal{S}}\dim E<\infty$. Indeed, in this case there is $G\in\mathcal{S}$ of maximal dimension and if $E\in\mathcal{S}$ is not included in $G$ then $E+G\in\mathcal{S}$ and $G\subsetneq E+G$ so $\dim G<\dim(E+G)$ which is not possible, hence $G$ is the greatest element of $\mathcal{S}$.

Let $\mathcal{S}_{\max}$ be the set of subspaces $E\in\mathcal{S}$ such that $E\neq\sup\mathcal{S}$ and there are no other elements of $\mathcal{S}$ between $E$ and $\sup\mathcal{S}$; in other terms, $\mathcal{S}_{\max}$ is the set of maximal elements of $\mathcal{S}\setminus\{\sup\mathcal{S}\}$. Then each element of $\mathcal{S}$ distinct of $\sup\mathcal{S}$ is majorated by an element of $\mathcal{S}_{\max}$ because if $F\in\mathcal{S},F\neq\sup\mathcal{S}$, among the elements $E\in\mathcal{S}$ which are $\neq\sup\mathcal{S}$ and contain $F$ there is one which has maximal dimension and this one belongs to $\mathcal{S}_{\max}$.

Clearly $\mathscr{C}(\sup\mathcal{S})$ is an ideal of $\mathscr{C}$. Let $\mathcal{P}\colon\mathscr{C}\to\mathscr{C}/\mathscr{C}(\sup\mathcal{S})$ be the canonical surjection of $\mathscr{C}$ onto the quotient $C^{*}$-algebra $\mathscr{C}/\mathscr{C}(\sup\mathcal{S})$. The main interest of the graded structure is that it gives an explicit description of $\mathscr{C}/\mathscr{C}(\sup\mathcal{S})$ and $\mathcal{P}$. Theorem A.4 is a consequence of the more general [21, Pr. 3.4]. If $\mathcal{S}$ is finite, which covers the N-body type situations, the proof is very easy [1, Th. 8.4.1].

If $\mathscr{A}_{i}$ are $C^{*}$-algebras then $\bigoplus_{i\in I}\mathscr{A}_{i}$ is the $C^{*}$-algebra consisting of families $A=(A_{i})_{i\in I}$ with $A_{i}\in\mathscr{A}_{i}$ satisfying $\|A\|\doteq\sup_{i}\|A_{i}\|<\infty$ and with the usual algebraic operations.

Theorem A.4**.**

If $S=\sup\mathcal{S}$ exists then the map

[TABLE]

is a morphism with $\ker\mathcal{P}=\mathscr{C}(\sup\mathcal{S})$. This gives a canonical embedding

[TABLE]

If $T\,\widetilde{\in}\,\mathscr{C}$ we define

[TABLE]

In later concrete situations $\mathscr{C}$ is realized on a Hilbert space $\mathcal{H}$ such that $\mathscr{C}(\sup\mathcal{S})=\mathscr{C}\cap K(\mathcal{H})$ (Lemma 3.6) and then $\mathrm{Sp_{ess}}(T)=\mathscr{C}\textnormal{-}\mathrm{Sp_{ess}}(T)$ for any $T\,\widetilde{\in}\,\mathscr{C}$ by Atkinson’s theorem [2, Th. 3.3.2] that we recall here. This gives a general version of the HVZ theorem.

Fix a Hilbert space $\mathcal{H}$. The quotient $C^{*}$-algebra $\mathcal{C}(\mathcal{H})\doteq B(\mathcal{H})/K(\mathcal{H})$ is the Calkin algebra of $\mathcal{H}$ and $\pi:B(\mathcal{H})\to\mathcal{C}(\mathcal{H})$ is the canonical morphism. Then for any $A\in B(\mathcal{H})$ we call localization at infinity of $A$ its image $\hat{A}\doteq\pi(A)$ in $\mathcal{C}(\mathcal{H})$; Atkinson’s theorem says that the essential spectrum of $A\in B(\mathcal{H})$ is the spectrum of $\hat{A}$:

[TABLE]

We extend this to observables as follows: if $A\colon C_{0}(\mathbb{R})\to B(\mathcal{H})$ is a morphism, hence an observable affiliated to $B(\mathcal{H})$, then $\hat{A}\doteq\pi\circ A$ is a morphism $C_{0}(\mathbb{R})\to\mathcal{C}(\mathcal{H})$, hence an observable affiliated to $\mathcal{C}(\mathcal{H})$, that we call localization at infinity of $A$, and we define the essential spectrum of $A$ as the spectrum of $\hat{A}$, so (A.7) remains valid. Thus $\mathrm{Sp_{ess}}(A)$ is the set of real $\lambda$ such that for any $\theta\in C_{0}(\mathbb{R})$ with $\theta(\lambda)\neq 0$ the operator $\theta(A)$ is not compact.

If $\mathscr{C}\subset B(\mathcal{H})$ is a $C^{*}$-subalgebra then $\mathscr{C}\cap K(\mathcal{H})$ is an ideal of $\mathscr{C}$ hence the quotient \widehat{\mathscr{C}\,}\doteq\mathscr{C}/\big{(}\mathscr{C}\cap K(\mathcal{H})\big{)} is a $C^{*}$-subalgebra of $\mathcal{C}(\mathcal{H})$ called localization at infinity of $\mathscr{C}$. If the observable $A$ is affiliated to $\mathscr{C}$ its localization at infinity $\widehat{A\,}$ is affiliated to $\widehat{\mathscr{C}\,}$. Thus

[TABLE]

because the restriction to $\mathscr{C}$ of the canonical morphism $B(\mathcal{H})\to\mathcal{C}(\mathcal{H})$ is injective.

A.3. Essential spectrum

Assume now $\Xi$ finite dimensional and let $\mathscr{C}$ be a $\mathbb{G}(\Xi)$-graded $C^{*}$-algebra. Since $\Xi$ is the maximal element of $\mathbb{G}(\Xi)$, if $T\,\widetilde{\in}\,\mathscr{C}$ then $\mathscr{C}\textnormal{-}\mathrm{Sp_{ess}}(T)=\mathrm{Sp}(\mathcal{P}T)$ is the spectrum of the image of $T$ in the quotient algebra $\mathscr{C}/\mathscr{C}(\Xi)$.

Theorem A.5**.**

If $T\in\mathscr{C}$ then $\{\mathcal{P}_{\scriptscriptstyle H}T\mid H\in\mathbb{H}(\Xi)\}$ is a compact subset of $\mathscr{C}$ and

[TABLE]

This relation is also valid for the observables affiliated to $\mathscr{C}$.

Theorem A.4 does not suffices to prove (A.9), the main new point is to show that the union in (A.9) is closed for all the elements of $\mathscr{C}$. The argument requires some preparations.

Lemma A.6**.**

Let $\mathscr{A}$ be a $C^{*}$-algebra, $I$ a set, and $\mathscr{A}^{[I]}$ the $C^{*}$-algebra of bounded functions $I\to\mathscr{A}$. If $A=(A_{i})_{i\in I}\in\mathscr{C}^{[I]}$ and $\mathcal{A}=\{A_{i}\mid i\in I\}$ is a compact subset of $\mathscr{A}$ then $\mathrm{Sp}(A)=\cup_{i\in I}\mathrm{Sp}(A_{i})$.

Proof.

We mays clearly assume that $\mathscr{A}$ is unital. If $\lambda\in\mathbb{C}$ then $A-\lambda=(A_{i}-\lambda)_{i\in I}$ is invertible in $\mathscr{A}^{[I]}$ if and only if each $A_{i}-\lambda$ is invertible in $\mathscr{A}$ and $\|(A_{i}-\lambda)^{-1}\|\leq C$ for some number $C$ independent of $i$. We have to prove that the last condition is automatically satisfied if $\mathcal{A}$ is compact. In this case $\mathcal{A}-\lambda$ is also compact so we may assume without loss of generality that $\lambda=0$. If $\mathscr{I}$ is the set of invertible elements of $\mathscr{A}$, then $\mathscr{I}$ is an open subset of $\mathscr{A}$ and the map $S\mapsto S^{-1}$ is continuous on $\mathscr{I}$. Since $\mathcal{A}$ is a compact subset of $\mathscr{I}$, we see that $\{A_{i}^{-1}\mid i\in I\}$ is a compact hence $\sup_{i}\|A_{i}^{-1}\|<\infty$. ∎

Lemma A.7**.**

$T\in\mathscr{C}$* and $E\in\mathbb{G}(\Xi)$, $E\neq\Xi\Rightarrow\exists H\in\mathbb{H}(\Xi)$ such that $\mathcal{P}_{\scriptscriptstyle E}T=\mathcal{P}_{\scriptscriptstyle H}T$.*

Proof.

We saw in Lemma A.1 that there is a countable subsemilattice $\mathcal{S}\subset\mathbb{G}(\Xi)$ such that $T\in\mathscr{C}(\mathcal{S})$. We will prove that there is $H\in\mathbb{H}(\Xi)$ such that $E\subset H$ but $F\not\subset H$ if $F\not\subset E$.

We have $F\not\subset E$ if and only if $E^{\perp}\not\subset F^{\perp}$ (orthogonals in $\Xi^{*}$) hence if and only if $E^{\perp}\cap F^{\perp}$ is a strict subspace of $E^{\perp}$. Thus $\{E^{\perp}\cap F^{\perp}\mid F\in\mathcal{S}_{\scriptscriptstyle E}^{\prime}\}$, cf. (A.1), is a countable set of strict subspaces of $E^{\perp}$ hence there is $L\in\mathbb{P}(\Xi^{*})$ such that $L\subset E^{\perp}$ and $L\not\subset F^{\perp}$ if $F\in\mathcal{S}_{\scriptscriptstyle E}^{\prime}$. Then $E\subset L^{\perp}$ and $F\not\subset L^{\perp}$ if $F\not\subset E$. Thus it suffices to take $H=L^{\perp}$.

Let $S\in\mathscr{C}(F)$ for some $F\in\mathcal{S}$. Then $\mathcal{P}_{\scriptscriptstyle E}S=S$ if $F\subset E$ and $\mathcal{P}_{\scriptscriptstyle E}S=0$ if $F\not\subset E$. In the first case we also have $F\subset H$ hence $\mathcal{P}_{\scriptscriptstyle H}S=S$ and in the second case $F\not\subset H$ hence $\mathcal{P}_{\scriptscriptstyle H}S=0$. Thus $\mathcal{P}_{\scriptscriptstyle E}=\mathcal{P}_{\scriptscriptstyle H}$ on each $\mathscr{C}(F)$ with $F\in\mathcal{S}$. Since these two operators are linear and continuous we get $\mathcal{P}_{\scriptscriptstyle E}=\mathcal{P}_{\scriptscriptstyle H}$ on $\mathscr{C}(\mathcal{S})$ and so $\mathcal{P}_{\scriptscriptstyle E}T=\mathcal{P}_{\scriptscriptstyle H}T$. ∎

Proof of Theorem A.5.

In this proof we abbreviate $\mathbb{H}=\mathbb{H}(\Xi)$. From Theorem A.4 it follows that $\mathscr{C}\textnormal{-}\mathrm{Sp_{ess}}(T)$ is the spectrum of the element $(\mathcal{P}_{\scriptscriptstyle E}T)_{E\in\mathbb{H}}$ of the $C^{*}$-algebra $\oplus_{E\in\mathbb{H}}\mathscr{C}_{\scriptscriptstyle E}$. Let $\mathscr{C}^{\scriptscriptstyle[\mathbb{H}]}$ be the $C^{*}$-algebra of bounded functions $\mathbb{H}\to\mathscr{C}$ with the $\sup$ norm. Then $\oplus_{E\in\mathbb{H}}\mathscr{C}_{\scriptscriptstyle E}$ is a $C^{*}$-subalgebra of $\mathscr{C}^{[\mathbb{H}]}$ hence for an element of $\oplus_{E\in\mathbb{H}}\mathscr{C}_{\scriptscriptstyle E}$ its spectrum in $\oplus_{E\in\mathbb{H}}\mathscr{C}_{\scriptscriptstyle E}$ coincides with its spectrum in $\mathscr{C}^{[\mathbb{H}]}$. Thus the spectrum of $(\mathcal{P}_{\scriptscriptstyle E}T)_{E\in\mathbb{H}}$ in $\oplus_{E\in\mathbb{H}}\mathscr{C}_{\scriptscriptstyle E}$ coincides with its spectrum in $\mathscr{C}^{[\mathbb{H}]}$. Thus by Lemma A.6 it suffices to show that $\mathcal{R}=\{\mathcal{P}_{\scriptscriptstyle E}T\mid E\in\mathbb{H}\}$ is a compact set in $\mathscr{C}$. By the last assertion of Theorem A.2 it suffices to show that $\mathcal{R}$ is norm closed. We will use the notation $P_{\scriptscriptstyle E}=\mathcal{P}_{\scriptscriptstyle E^{\perp}}$ for $E\in\mathbb{G}(\Xi^{*})$ hence $\mathcal{R}=\{P_{\scriptscriptstyle L}T\mid L\in\mathbb{P}\}$ where $\mathbb{P}\equiv\mathbb{P}(\Xi^{*})$.

We make two remarks concerning the $P_{\scriptscriptstyle E}$. If $E,F\in\mathbb{G}(\Xi^{*})$ then by (A.3)

[TABLE]

Then if $P_{\scriptscriptstyle E}^{\prime}=1-P_{\scriptscriptstyle E}$ we have $P_{\scriptscriptstyle E}-P_{\scriptscriptstyle F}=P_{\scriptscriptstyle E}P_{\scriptscriptstyle F}^{\prime}-P_{\scriptscriptstyle E}^{\prime}P_{\scriptscriptstyle F}$ hence $P_{\scriptscriptstyle E}(P_{\scriptscriptstyle E}-P_{\scriptscriptstyle F})=P_{\scriptscriptstyle E}P_{\scriptscriptstyle F}^{\prime}$, which gives for all $S\in\mathscr{C}$

[TABLE]

Let $S$ be an accumulation point of $\mathcal{R}$. Then there is a sequence of elements $L_{n}\in\mathbb{P}$ with $L_{n}\neq L_{m}$ if $n\neq m$ such that $\|P_{\scriptscriptstyle L_{n}}T-S\|\rightarrow 0$. The sequence of subspaces $E_{n}=\sum_{m\geq n}L_{m}\neq 0$ is decreasing hence there is a subspace $E$ of dimension $k>0$ such that $E_{n}=E$ for large $n$. Then for each such $n$ one can find integers $n_{1}<n_{2}<\dots<n_{k}$ with $n_{1}=n$ such that $E=L_{n_{1}}+\dots+L_{n_{k}}$. If $\varepsilon>0$ then there is $N$ such that $\|P_{\scriptscriptstyle L_{n}}T-P_{\scriptscriptstyle L_{m}}T\|\leq\varepsilon$ if $n,m\geq N$. Choose a sequence $n_{1}=n<n_{2}<\dots<n_{k}$ as above and denote $F_{i}=L_{n_{1}}+\dots+L_{n_{i-1}}$. The relations (A.11) and (A.10) imply

[TABLE]

On the other hand, by using (A.10) again we can write

[TABLE]

Repeating this procedure we get

[TABLE]

Thus we have $\left\|P_{\scriptscriptstyle L_{n}}T-P_{\scriptscriptstyle E}T\right\|\leq(k-1)\varepsilon$. This shows that $S=P_{\scriptscriptstyle E}T=\mathcal{P}_{\scriptscriptstyle E^{\perp}}T$ with $E^{\perp}\neq\Xi$ and now it suffices to use Lemma A.7 with $E$ replaced by $E^{\perp}$.

The extension to observables is straightforward by using the spectral mapping theorem. If $z$ is a nonreal number and $R_{T}(z)$ is the resolvent of the observable $T$, by taking into account the relation $\mathcal{P}_{\scriptscriptstyle H}R_{T}(z)=R_{\mathcal{P}_{\scriptscriptstyle H}T}(z)$ due to the fact that $\mathcal{P}_{\scriptscriptstyle H}$ is a morphism, it suffices to use (A.9) with $T$ replaced by $R_{T}(z)$. ∎

Theorem A.8**.**

If $\mathcal{S}\subset\mathbb{G}(\Xi)$ is a subsemilattice with $\Xi\in\mathcal{S}$ and $T\,\widetilde{\in}\,\mathscr{C}(\mathcal{S})$ then

[TABLE]

Proof.

Since $\mathscr{C}(\mathcal{S})/\mathscr{C}(\Xi)$ is a $C^{*}$-subalgebra of $\mathscr{C}/\mathscr{C}(\Xi)$ the spectrum of the image of $T$ in the quotient algebra $\mathscr{C}(\mathcal{S})/\mathscr{C}(\Xi)$ is the same as the spectrum of the image of $T$ in the quotient algebra $\mathscr{C}/\mathscr{C}(\Xi)$ hence is given by the formula (A.9). If $E\neq\Xi$ there is $H\in\mathbb{H}(\Xi)$ such that $E\subset H$ and then $\mathcal{P}_{\scriptscriptstyle E}=\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle H}$ by (A.3) hence $\mathcal{P}_{\scriptscriptstyle E}T=\mathcal{P}_{\scriptscriptstyle E}\mathcal{P}_{\scriptscriptstyle H}T$ and since $\mathcal{P}_{\scriptscriptstyle E}$ is a morphism we get $\mathrm{Sp}(\mathcal{P}_{\scriptscriptstyle E}T)\subset\mathrm{Sp}(\mathcal{P}_{\scriptscriptstyle H}T)$. Thus

[TABLE]

Observe that the preceding argument gives $\mathrm{Sp}(\mathcal{P}_{\scriptscriptstyle F}T)\subset\mathrm{Sp}(\mathcal{P}_{\scriptscriptstyle E}T)$ if $F\subset E$ hence

[TABLE]

i.e. only the largest spaces $E$ are significant in the union. It remains to show that for any $T$ and any hyperplane $H$ there is $E\in\mathcal{S}$ such that $E\subset H$ and $\mathrm{Sp}(\mathcal{P}_{\scriptscriptstyle E}T)=\mathrm{Sp}(\mathcal{P}_{\scriptscriptstyle H}T)$. Let $\mathcal{S}_{H}$ be the set of $F\in\mathcal{S}$ such that $F\subset H$ and choose $E\in\mathcal{S}_{H}$ of maximal dimension. If $F\in\mathcal{S}_{H}$ then $E+F\in\mathcal{S}_{H}$ and if $F\not\subset E$ then $\dim(E+F)>\dim E$ which is impossible, hence $F\subset E$. Thus $E$ is the greatest element of the subsemilattice $\mathcal{S}_{H}$. Then we clearly have $\mathcal{P}_{\scriptscriptstyle E}T=\mathcal{P}_{\scriptscriptstyle H}T$ for any $T\in\mathring{\mathscr{C}}(\mathcal{S})$ hence by continuity for any $T\in\mathscr{C}(\mathcal{S})$. ∎

A.4. Affiliation criteria

We first recall the notion of sum in form sense. Let $H_{0}$ be a self-adjoint operator on $\mathcal{H}$ and $\mathcal{G}=D(|H_{0}|^{\frac{1}{2}})$ its form domain with the graph topology. If $\mathcal{G}^{*}$ is the adjoint of $\mathcal{G}$ (space of continuous anti-linear forms) and if we identify $\mathcal{H}^{*}=\mathcal{H}$ via Riesz Lemma, we get continuous dense embeddings $\mathcal{G}\subset\mathcal{H}\subset\mathcal{G}^{*}$ and $H_{0}$ extends to a continuous map $\mathcal{G}\to\mathcal{G}^{*}$ for which we keep the notation $H_{0}$. A symmetric operator $V\colon\mathcal{G}\to\mathcal{G}^{*}$ is a form perturbation of $H_{0}$ if there are numbers $\mu,\nu\geq 0$ with $\mu<1$ such that either $\pm V\leq\mu|H_{0}|+\nu$ or $H_{0}$ is bounded from below and $V\geq-\mu H_{0}-\nu$. Then the restriction of $H=H_{0}+V\colon\mathcal{G}\to\mathcal{G}^{*}$ to $D(H)\doteq\{g\in\mathcal{G}\mid Hg\in\mathcal{H}\}$ is a self-adjoint operator on $\mathcal{H}$, still denoted $H$, called form sum of $H_{0}$ and $V$. By [21, Th.​ 2.8, Lm.​ 2.9]:

Theorem A.9**.**

Assume that $H_{0}$ is strictly affiliated to a $C^{*}$-algebra $\mathscr{C}\subset B(\mathcal{H})$. If $V$ is a form perturbation of $H_{0}$ such that $\varphi(H_{0})V(|H_{0}|+1)^{-1/2}\in\mathscr{C}$ for all $\varphi\in C_{\mathrm{c}}(\mathbb{R})$ then the form sum $H=H_{0}+V$ is strictly affiliated to $\mathscr{C}$.

Let $\Xi$ be an arbitrary real vector space, $\mathcal{S}\subset\mathbb{G}(\Xi)$ a subsemilattice with a least element $X$, and $\mathscr{C}$ an $\mathcal{S}$-graded $C^{*}$-algebra of operators on a Hilbert space $\mathcal{H}$. We give here methods for constructing self-adjoint operators affiliated to $\mathscr{C}$. Note that

[TABLE]

Proposition A.10 is very easy but interesting because $V$ is not required to be decomposable.

Proposition A.10**.**

Let $H_{\scriptscriptstyle{X}}$ be a self-adjoint operator on $\mathcal{H}$ affiliated to $\mathscr{C}(X)$. If $V\in\mathscr{C}$ is symmetric then $H\doteq H_{\scriptscriptstyle{X}}+V$ is affiliated to $\mathscr{C}$ and $\mathcal{P}_{\scriptscriptstyle E}H=H_{\scriptscriptstyle{X}}+\mathcal{P}_{\scriptscriptstyle E}V$.

Proof.

If $z\in\mathbb{C}$ we set $R_{\scriptscriptstyle{X}}(z)=(z-H_{\scriptscriptstyle{X}})^{-1}$ and $R(z)=(z-H)^{-1}$ whenever the inverses exist. If $\Im z$ is large enough then $\|VR_{\scriptscriptstyle{X}}(z)\|<1$ and then, by using the relation $z-H=(1-VR_{\scriptscriptstyle{X}}(z))(z-H_{\scriptscriptstyle{X}})$, we see that $z-H$ is invertible and

[TABLE]

We have $R_{\scriptscriptstyle{X}}(z)\in\mathscr{C}(X)$ hence $VR_{\scriptscriptstyle{X}}(z)\in\mathscr{C}$ and since $\mathcal{P}_{\scriptscriptstyle E}:\mathscr{C}\to\mathscr{C}_{\scriptscriptstyle E}$ is a morphism

[TABLE]

where $V_{\scriptscriptstyle E}=\mathcal{P}_{\scriptscriptstyle E}V$. The series in (A.13) is norm convergent hence $R(z)\in\mathscr{C}$ and

[TABLE]

Since $\|V_{\scriptscriptstyle E}R_{\scriptscriptstyle{X}}(z)\|=\|\mathcal{P}_{\scriptscriptstyle E}[VR_{\scriptscriptstyle{X}}(z)]\|\leq\|VR_{\scriptscriptstyle{X}}(z)\|$ the series is norm convergent. ∎

Now we consider unbounded $V$. Let $H_{\scriptscriptstyle{X}}$ be a positive self-adjoint operator on $\mathcal{H}$ with form domain $\mathcal{G}$. For each $E\in\mathcal{S}$ let $V(E):\mathcal{G}\to\mathcal{G}^{*}$ symmetric such that $V(X)=0$ and

(1) the family $\{V(E)\}_{E\in\mathcal{S}}$ is norm summable in $B(\mathcal{G},\mathcal{G}^{*})$;

(2) $V(E)\geq-\mu_{\scriptscriptstyle E}H_{\scriptscriptstyle{X}}-\nu_{\scriptscriptstyle E}$ with $\mu_{\scriptscriptstyle E},\nu_{\scriptscriptstyle E}\geq 0$, $\sum_{E\in\mathcal{S}}\mu_{\scriptscriptstyle E}<1$, and $\sum_{E\in\mathcal{S}}\nu_{\scriptscriptstyle E}<\infty$.

Denote $V=\sum_{E\in\mathcal{S}}V(E)$ and $V_{\scriptscriptstyle E}=\sum_{{\scriptscriptstyle F}\leq{\scriptscriptstyle E}}V(F)$. Then the form sums $H=H_{\scriptscriptstyle{X}}+V$ and $H_{\scriptscriptstyle E}=H_{\scriptscriptstyle{X}}+V_{\scriptscriptstyle E}$ are bounded from below self-adjoint operators on $\mathcal{H}$ with form domain equal to $\mathcal{G}$. Then by using Lemma 2.9 and Theorem 3.5 from [21], we get

Theorem A.11**.**

Assume $\mathscr{C}(X)\cdot\mathscr{C}(E)=\mathscr{C}(E)\ \forall E\in\mathcal{S}$. If $H_{\scriptscriptstyle{X}}$ is strictly affiliated to $\mathscr{C}(X)$ and

[TABLE]

then $H$ is strictly affiliated to $\mathscr{C}$ and $\mathcal{P}_{\scriptscriptstyle E}H=H_{\scriptscriptstyle E}$ for all $E\in\mathcal{S}$.