This paper extends the concept of continuous calculus from C*-algebras to a broader class of Su*-algebras, enabling functional calculus for unbounded elements without relying on representation theory.
Contribution
It introduces the existence of universal continuous calculi for finite tuples of commuting Hermitian elements in Su*-algebras, generalizing known results for C*-algebras.
Findings
01
Established universal continuous calculus for Su*-algebras
02
Derived new results on *-algebras of continuous functions
03
Provided an elementary approach avoiding representation theory
Abstract
Universal continuous calculi are defined and it is shown that for every finite tuple of pairwise commuting Hermitian elements of a Su*-algebra (an ordered *-algebra that is symmetric, i.e. "strictly" positive elements are invertible, and uniformly complete), such a universal continuous calculus exists. This generalizes the continuous calculus for C*-algebras to a class of generally unbounded ordered *-algebras. On the way, some results about *-algebras of continuous functions on locally compact spaces are obtained. The approach used throughout is rather elementary and especially avoids any representation theory.
\displaystyle\mathcal{I}_{\mathbbm{R}}\coloneqq\big{\{}\,f\in\mathscr{C}(\mathbbm{R}^{N})\;\big{|}\;\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}\big{)}\textup{ is invertible in }\mathcal{A}\,\big{\}}
\displaystyle\mathcal{I}_{\mathbbm{R}}\coloneqq\big{\{}\,f\in\mathscr{C}(\mathbbm{R}^{N})\;\big{|}\;\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}\big{)}\textup{ is invertible in }\mathcal{A}\,\big{\}}
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TopicsAdvanced Operator Algebra Research · Advanced Algebra and Logic · Mathematical Analysis and Transform Methods
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Universal Continuous Calculus for Su∗/̄Algebras
Matthias Schötz
Département de Mathématiques
Université libre de Bruxelles
Boursier de l’ULB, [email protected].
This work was supported by the Fonds de la Recherche Scientifique (FNRS) and the Fonds Wetenschappelijk
Onderzoek - Vlaaderen (FWO) under EOS Project n030950721.
(November 2020)
Abstract
Universal continuous calculi are defined and it is shown that for every finite tuple
of pairwise commuting Hermitian elements of a Su∗/̄algebra
(an ordered ∗/̄algebra that is symmetric, i.e. “strictly” positive elements are
invertible, and uniformly complete), such a universal continuous calculus exists. This generalizes
the continuous calculus for C∗/̄algebras to a class of generally unbounded ordered ∗/̄algebras.
On the way, some results about ∗/̄algebras of continuous functions on locally
compact spaces are obtained. The approach used throughout is rather elementary
and especially avoids any representation theory.
1 Introduction
The term ∗/̄algebra will always refer to a unital associative (but not necessarily commutative)
∗/̄algebra over the scalar field of complex numbers. Su∗/̄algebras have been introduced
and examined in [14].
It was shown that they preserve many basic properties of C∗/̄algebras, but may have unbounded elements,
and that the commutative Su∗/̄algebras are complexifications of complete Φ/̄algebras.
Essentially, they are ∗/̄algebras endowed with a partial order on the Hermitian elements,
which are complete with respect to a metric topology induced by the order and fulfil one of
several equivalent additional conditions like existence of absolute values, of square roots of positive elements,
of certain finite suprema or infima, or of inverses of Hermitian elements that are “strictly” positive.
Like for C∗/̄algebras, one can now ask whether these constructions can be generalized
to a well-defined way to apply more or less arbitrary continuous functions to algebra elements.
More precisely, given N∈\mathbbmN and pairwise commuting Hermitian elements a1,…,aN of a
Su∗/̄algebra A, does there
exist a unital ∗/̄homomorphism Φ from a ∗/̄algebra I of continuous
complex-valued functions on
\mathbbmRN, or on a closed subset of \mathbbmRN, to A, that maps the N coordinate functions
to a1,…,aN? If N=1 and a:=a1, then Φ would map
polynomial functions to the corresponding polynomials of a. Similarly, one would expect
that e.g. the absolute value function ∣\ignorespaces⋅\ignorespaces∣:\mathbbmR→[0,∞[ is mapped to the
absolute value of a, or, if a is positive, the square root function
\ignorespaces⋅\ignorespaces:[0,∞[→[0,∞[ should be mapped to the square root of a.
Likewise, for N=2, the function max:\mathbbmR2→\mathbbmR, (x1,x2)↦max{x1,x2}
should be mapped to the supremum a1∨a2. As a continuous calculus necessarily
maps to a commutative unital ∗/̄subalgebra of A, the continuous calculus for
Φ/̄algebras that has been constructed in [2]
already provides a solution for this in some cases.
Aside from the existence of such a unital ∗/̄homomorphism Φ, one would also
want Φ to be uniquely determined. Once the ∗/̄algebra of functions
I is fixed, uniqueness of Φ can typically be obtained e.g. from the
generalized Stone-Weierstraß Theorem of [15].
But finding a good choice for I is not easy: Following
[2, Thm. 4.12],
the algebra I can be chosen as the ∗/̄algebra of all polynomially
bounded continuous functions on \mathbbmRN and it was shown that for this choice, Φ is uniquely determined.
Yet this way, neither the square root function \ignorespaces⋅\ignorespaces:[0,∞[→[0,∞[
nor the exponential function exp:\mathbbmR→\mathbbmR are elements of I in the one-dimensional
case.
While one might not be able to construct continuous calculi for all Hermitian elements a
of A that allow to give a and exp(a) a well-defined meaning (e.g. if
a is not positive in the first case, or if a is “not bounded enough” in some sense to let exp(a)
be an element of A) there certainly are cases where this is possible. In
general, there might be several different continuous calculi for the same N/̄tuple
a1,…,aN, that differ in the choice of the algebra of functions on which
they are defined.
For C∗/̄algebras, there exists an easy natural choice for I at least in
the one-dimensional case: One defines I to be
the C∗/̄algebra of all continuous functions on the compact spectrum of a.
This allows the application of all continuous functions with domain of definition
larger than spec(a), especially also the application of the square root function if
a is positive, which is the case if and only if spec(a)⊆[0,∞[.
It will be shown that this natural choice is essentially also viable for Su∗/̄algebras,
but there arises an additional problem: As elements of Su∗/̄algebras
may be unbounded, the spectrum spec(a) is no longer necessarily compact,
hence there typically are several different ∗/̄algebras in between
the ∗/̄algebra \mathscr{C}\big{(}\operatorname{\mathrm{spec}}(a)\big{)} of all continuous functions and
its ∗/̄subalgebra of polynomially bounded ones.
The best choice will of course be the largest possible one.
This leads to the concept of a universal continuous calculus
in Definition 3.5:
There might be many continuous calculi for the same tuple a1,…,aN,
and the universal continuous calculus (if it exists) is the one through which all
others can be factored in a natural way. Consequently, the universal continuous
calculus is the most general one, and the algebra of functions on which it is
defined carries information about the elements a1,…,aN.
The main result, Theorem 5.11, establishes the existence of such a
universal continuous calculus for all N/̄tuples of pairwise commuting Hermitian
elements of a Su∗/̄algebra. This will be proven by a rather explicit
construction using elementary methods. Moreover, it will be shown that the universal
continuous calculus is an order embedding of a Su∗/̄algebra of continuous functions
into a general Su∗/̄algebra.
This allows to prove identities or estimates involving finitely
many pairwise commuting Hermitian elements and their absolute values, square roots, etc.,
by discussing only the special and easy case that they are given by continuous functions.
It also yields a representation of certain Su∗/̄algebras for which a surjective
continuous calculus exists.
This is roughly analogous to the result provided by [3]
for Riesz spaces. Note, however, that the approach taken here is quite different:
While [2] applies the
representation theorem obtained in [3] for
“small” Riesz spaces in order to derive a functional calculus, the functional
calculus for Su∗/̄algebras here will be obtained by extending the polynomial
calculus and can then be used to obtain a representation theorem for “small” commutative Su∗/̄algebras.
The article is organized as follows: After recapitulating some preliminaries and fixing the
notation in the next Section 2, Section 3 discusses
the fundamental definitions that are relevant for this article, especially (universal)
continuous calculi and commutative Su∗/̄algebras of finite type.
In Section 4, certain well-behaved commutative Su∗/̄algebras of
continuous functions will be examined in some detail.
The construction of universal continuous calculi is presented in Section 5
and results in the main Theorem 5.11. An application of this is a representation
theorem for commutative Su∗/̄algebras of finite type, which is discussed in the final
Section 6 and which gives an algebraic characterization of those ordered ∗/̄algebras
of continuous functions on a closed subset of \mathbbmRN that contain all uniformly bounded
continuous functions and at least one proper function.
2 Preliminaries
The notation essentially follows [14],
where the fundamental properties of ordered ∗/̄algebras
and especially Su∗/̄algebras have been discussed in more detail and more generality.
Natural numbers are \mathbbmN={1,2,3,…} and \mathbbmN0:=\mathbbmN∪{0}, and
the fields of real and complex numbers are denoted by \mathbbmR and \mathbbmC, respectively.
Let X be a set, then idX:X→X is x↦idX(x):=x. A partial order
on X is a reflexive, transitive and anti-symmetric relation. If X and Y are both
endowed with a partial order, then a map Φ:X→Y is called increasing
if Φ(x)≤Φ(x′) for all x,x′∈X with x≤x′. If Φ is
injective, increasing and also fulfils x≤x′ for all x,x′∈X with Φ(x)≤Φ(x′),
then Φ is called an order embedding.
A ∗/̄algebra is a unital associative algebra over the field \mathbbmC that
is endowed with an anti-linear involution \ignorespaces⋅\ignorespaces∗ fulfilling (ab)∗=b∗a∗
for all its elements a and b. The unit of a ∗/̄algebra A is denoted by \mathbbm1, or, more explicitly,
by \mathbbm1A, and automatically fulfils \mathbbm1∗=\mathbbm1. Note that it is not
required that 0=\mathbbm1, but the only case where 0=\mathbbm1 is
the trivial ∗/̄algebra {0}. An element a∈A is called Hermitian
if a=a∗, and AH:={a∈A∣a=a∗} denotes the
real linear subspace of Hermitian elements of A.
Every a∈A can be decomposed as a=Re(a)+iIm(a) with uniquely determined
Hermitian real and imaginary parts of a, which are explicitly given by Re(a)=21(a+a∗) and Im(a)=2i1(a−a∗).
Moreover, if S⊆A is stable under the ∗/̄involution, then its commutant
S^{\prime}\coloneqq\big{\{}\,a\in\mathcal{A}\;\big{|}\;\forall_{s\in S}:as=sa\,\big{\}}
and its bicommutant S′′ are unital ∗/̄subalgebras of A, i.e.
unital subalgebras that are stable under the ∗/̄involution and thus are ∗/̄algebras again.
Note that S′′ is commutative if all elements in S are pairwise commuting.
The unital ∗/̄subalgebra generated by a subset S of A is the smallest (with respect to ⊆)
unital ∗/̄subalgebra of A that contains S, and can explicitly be constructed as the set
of all finite sums of finite products of elements of S∪{s∗∣s∈S}∪{λ\mathbbm1∣λ∈\mathbbmC}.
A linear subspace I of a ∗/̄algebra A is called a ∗/̄ideal if it is stable
under the ∗/̄involution and fulfils ab∈I for all a∈A and all b∈I,
and then automatically also fulfils ba∈I for all a∈A and all b∈I.
Given two ∗/̄algebras A and B, then a map Φ:A→B
is a unital ∗/̄homomorphism if it is linear, maps \mathbbm1A to \mathbbm1B and fulfils Φ(aa~)=Φ(a)Φ(a~)
and Φ(a∗)=Φ(a)∗ for all a,a~∈A. Its kernel kerΦ:={a∈A∣Φ(a)=0}
automatically is a ∗/̄ideal. As an example, \mathbbmC[t1,…,tN] denotes the ∗/̄algebra of polynomials in N∈\mathbbmN Hermitian arguments,
i.e. its ∗/̄involution is given by complex conjugation of the coefficients so that \mathbbmC[t1,…,tN]H≅\mathbbmR[t1,…,tN]. This is the
∗/̄algebra that is freely generated by N pairwise commuting Hermitian elements t1,…,tN,
so for every ∗/̄algebra A and every N-tuple of pairwise commuting a1,…,aN∈AH
there exists a unique unital ∗/̄homomorphism from \mathbbmC[t1,…,tN] to A that maps
tn to an for all n∈{1,…,N}. This unital ∗/̄homomorphism is denoted as usual as evaluating a
polynomial on a1,…,aN, i.e. as \mathbbmC[t1,…,tN]∋π↦π(a1,…,aN)∈A.
The idea of a functional calculus is to extend this “polynomial calculus” to more interesting ∗/̄algebras like C(\mathbbmRN),
the continuous complex valued functions on \mathbbmRN with the pointwise operations.
An ordered ∗/̄algebras is a ∗/̄algebra A that carries a partial order ≤ on AH such that
[TABLE]
hold for all a,b,c∈AH with a≤b and all d∈A. One writes AH+:={a∈AH+∣a≥0}
for the convex cone of positive Hermitian elements of A, and it is not hard to check that the order ≤ on AH
is completely determined by AH+. As \mathbbm1∈AH+, a+b∈AH+ for all a,b∈AH+
and c∗ac∈AH for all a∈AH+ and all c∈A, this set AH+ is a quadratic module.
A unital ∗/̄subalgebra B of an ordered ∗/̄algebra A will always be endowed with the order that BH
inherits from AH.
If A and B are two ordered ∗/̄algebras and Φ:A→B
a unital ∗/̄homomorphism, then Φ is called positive if its restriction to a map from AH to BH
is increasing, or equivalently, if Φ(a)∈BH+ for all a∈AH+.
For example, the ∗/̄algebra \mathbbmC[t1,…,tN] becomes an ordered ∗/̄algebra by endowing \mathbbmC[t1,…,tN]H with the pointwise order,
which is the order for which \mathbbmC[t1,…,tN]H+
is the set of all π∈\mathbbmC[t1,…,tN]H that fulfil π(x1,…,xN)≥0 for all x∈\mathbbmRN.
One might ask now whether the canonical unital ∗/̄homomorphism \mathbbmC[t1,…,tN]∋π↦π(a1,…,aN)∈A
is positive for every choice of pairwise commuting Hermitian elements a1,…,aN of any ordered ∗/̄algebra A
and with respect to the pointwise order on \mathbbmC[t1,…,tN]. While this can be shown to be true in the one-dimensional case
by expressing every π∈\mathbbmC[t]H+ as a sum of polynomials of the form ρ∗ρ with ρ∈\mathbbmC[t]
(which is possible as a consequence of the fundamental theorem of algebra), it does no longer hold if N≥2:
Indeed, the identity map id\mathbbmC[t1,…,tN]
is not positive as a unital ∗/̄homomorphism from \mathbbmC[t1,…,tN] with the pointwise order
to \mathbbmC[t1,…,tN] with the order for which only elements of the form ∑k=1Kρk∗ρk with K∈\mathbbmN and
ρ1,…,ρK∈\mathbbmC[t1,…,tN] are positive: For example, the famous Motzkin polynomial
t14t22+t12t24−3t12t22+1 is pointwise positive but cannot be expressed as a sum of squares.
This already indicates that the theory of quadratic modules on polynomial algebras in real algebraic geometry
is highly non-trivial. Important results are the Positivstellensatz by Krivine and Stengle,
[10] and [16], which gives an algebraic
description of polynomials that are pointwise positive or strictly positive on a semi-algebraic set,
and its variants in compact cases by Handelman [6],
Schmüdgen [13], and Putinar [11].
At least in some special cases, such algebraic descriptions can also be obtained constructively by
exploiting a classical result of Pólya, see e.g. [1].
In a similar way, [5] provides a constructive solution
to Hilbert’s 17th problem in the special case of strictly positive polynomials. These constructive
methods will be helpful later on in the proof of Lemma 5.1 for
extending the polynomial calculus to a continuous calculus without having to resort to the full might
of the Positivstellensatz.
Another important example of ordered ∗/̄algebras is C(X), the ∗/̄algebra
of complex-valued continuous functions on a topological space X with the
pointwise operations and pointwise comparison (if X=∅, then C(X)≅{0}). Similarly, every C∗/̄algebra
becomes an ordered ∗/̄algebra in a natural way by declaring those Hermitian elements
to be positive whose spectrum is a subset of [0,∞[. These are two
classes of examples of Su∗/̄algebras, which will be discussed
in the following (see [14] for details):
An ordered ∗/̄algebra A is called Archimedean if AH is an Archimedean
ordered vector space, i.e. if the following condition is fulfilled: Whenever a≤ϵb
holds for one a∈AH, one b∈AH+ and all ϵ∈]0,∞[,
then a≤0. It is important to point out that this is not related to the notion of an Archimedean
quadratic module in real algebraic geometry.
Using the convention that −∞\mathbbm1≤a≤∞\mathbbm1 and a≤∞2\mathbbm1 are true for
all a∈AH, one can define on every Archimedean ordered ∗/̄algebra A the map
∥\ignorespaces⋅\ignorespaces∥∞:A→[0,∞],
[TABLE]
and the subset \mathcal{A}^{\mathrm{bd}}\coloneqq\big{\{}\,a\in\mathcal{A}\;\big{|}\;\lVert a\rVert_{\infty}<\infty\,\big{\}} of uniformly bounded elements of A.
Alternatively,
[TABLE]
holds for all a∈AH. Then Abd is a unital ∗/̄subalgebra of A on which the restriction
of ∥\ignorespaces⋅\ignorespaces∥∞ is a C∗/̄norm, i.e. a norm
fulfilling ∥ab∥∞≤∥a∥∞∥b∥∞ and ∥a∗a∥∞=∥a∥∞2
for all a,b∈Abd. This relation between Archimedean ordered ∗/̄algebras and C∗/̄algebras
has been described first in [4],
and might serve as a motivation to study Archimedean ordered ∗/̄algebras as abstractions
of ∗/̄algebras of possibly unbounded operators on a Hilbert space, i.e. of O∗/̄algebras as in [12].
In the unbounded case,
if Abd=A, the map ∥\ignorespaces⋅\ignorespaces∥∞ does not describe a norm on A
but can still be used to construct a translation-invariant
metric d∞ on A, called the uniform metric, as
[TABLE]
for all a,b∈A. An Archimedean ordered ∗/̄algebra is called uniformly complete if it
is complete with respect to d∞. All topological and metric notions for
Archimedean ordered ∗/̄algebras will always refer to this uniform metric. For example,
every positive unital ∗/̄homomorphism between two Archimedean ordered ∗/̄algebras is
automatically continuous.
If A is an ordered ∗/̄algebra, then an element a∈AH is called coercive
if there exists an ϵ∈]0,∞[ such that ϵ\mathbbm1≤a,
thus coercive elements are especially positive. If the multiplicative inverse a−1 of this coercive a exists in A, then
it is Hermitian and 0≤a−1≤ϵ−1\mathbbm1.
An ordered ∗/̄algebra in which all coercive elements are invertible
is called symmetric. A Su∗-algebra finally is a symmetric and uniformly complete Archimedean ordered ∗/̄algebra.
Examples are all C∗/̄algebras with the canonical order (these are the Su∗/̄algebras A for which A=Abd)
as well as complexifications of complete Φ/̄algebras (which are the commutative Su∗/̄algebras),
especially C(X) for every topological space X.
See e.g. [8] for more details
on Φ/̄algebras. One can check that ∥f∥∞=supx∈X∣f(x)∣
holds for all f∈C(X) and all topological spaces X, and it is worth mentioning
that the order on C(X)H is usually easier to work with than the axioms (2.1).
Given a continuous map ϕ:X→Y between two topological spaces X and Y,
then one obtains a positive unital ∗/̄homomorphism C(Y)∋f↦f∘ϕ∈C(X).
In the special case that X is a subset of Y and ϕ the inclusion map, then f∘ϕ
is simply the restriction of f∈C(Y) to X, denoted by f∣X∈C(X).
More examples of Su∗/̄algebras can be constructed as ∗/̄algebras of unbounded operators on a Hilbert space
if some selfadjointness conditions are fulfilled that guarantee the existence
of inverses of coercive elements. Su∗/̄algebras have some nice properties, for example, one can
construct square roots of positive Hermitian elements, or absolute values of Hermitian elements,
and also adapted versions of infima and suprema of finitely many commuting Hermitian elements.
However, for the purpose of this article, these constructions are only needed in the special case
of C(X), where they simply describe the pointwise square root, pointwise absolute value, and pointwise
minimum or maximum of real-valued functions. The universal continuous calculus then provides
an equivalent way to generalize these constructions to arbitrary Su∗/̄algebras.
It is also noteworthy that on a Su∗/̄algebra A, in contrast to the case of ordered ∗/̄algebras
of polynomials, the order is uniquely determined in the sense that there exists only one partial
order on the Hermitian elements of the underlying ∗/̄algebra that fulfils the axioms (2.1).
3 Universal Continuous Calculus – Definitions
Consider a Su∗/̄algebra A. Then every unital ∗/̄subalgebra B of A
with the order inherited from A is again an Archimedean ordered ∗/̄algebra. It is
even a Su∗/̄algebra if and only if it is symmetric itself and closed with respect to the uniform
metric d∞. One obvious example for this is B=Abd, the C∗-algebra of all
uniformly bounded elements of A. However, there arise some difficulties
when one attempts to construct other examples of closed symmetric unital ∗/̄subalgebras: The naive approach
to start with a symmetric unital ∗/̄subalgebra and take its closure with respect to d∞
might fail because left and right multiplication with elements of A is in general not continuous with
respect to d∞, which makes it harder to guarantee that the closure is again a subalgebra.
Nevertheless, there are important special cases which are easier to understand:
Definition 3.1**.**
*Let A be an ordered ∗/̄algebra. Then an intermediate ∗/̄subalgebra of A
is a (necessarily unital) ∗/̄subalgebra I of A fulfilling
Abd⊆I.
*
Proposition 3.2**.**
*Let A be a Su∗/̄algebra and I an intermediate ∗/̄subalgebra
of A, then I is a closed unital ∗/̄subalgebra of A and symmetric,
hence is again a Su∗/̄algebra. Moreover, whenever
ℓ,u∈IH and a∈AH fulfil ℓ≤a≤u,
then also a∈IH.
*
Proof**:**
As the inverse of every coercive element in IH is uniformly bounded,
I is again symmetric.
Moreover, if a∈AH and ℓ,u∈IH
fulfil ℓ≤a≤u, define
b:=21(\mathbbm1−ℓ)2+21(\mathbbm1+u)2+\mathbbm1∈IH+.
Then −b≤−21(\mathbbm1−ℓ)2≤ℓ and u≤21(\mathbbm1+u)2≤b
imply −b≤a≤b. As b is coercive with b≥\mathbbm1 it follows that
b−1∈(Abd)H+ exists and that
−b−1≤b−1ab−1≤b−1.
So b−1ab−1∈(Abd)H⊆IH
and therefore also a∈IH.
It only remains to show that I is closed in A, and as the \mathbbmR-linear projections
A∋a↦Re(a)=(a+a∗)/2∈AH and A∋a↦Im(a)=(a−a∗)/(2i)∈AH
onto the real and imaginary parts are continuous by continuity of addition, ∗/̄involution and scalar multiplication,
it suffices to check that IH is closed in AH: If a sequence (an)n∈\mathbbmN
in IH converges against a limit a^∈AH, then there exists an n∈\mathbbmN
such that −\mathbbm1≤a^−an≤\mathbbm1, i.e. an−\mathbbm1≤a^≤an+\mathbbm1,
which implies a^∈IH.
\boxempty
A continuous calculus should especially generalize the polynomial calculus. The continuous functions on \mathbbmRN
that correspond to the generators t1,…,tN of the ∗/̄algebra \mathbbmC[t1,…,tN] are the coordinate
functions:
Definition 3.3**.**
*If X is a closed subset of \mathbbmRN with N∈\mathbbmN, then the coordinate functions on X
are defined as prX;n:X→\mathbbmR, (x1,…,xN)↦prX;n(x1,…,xN):=xn
for all n∈{1,…,N}. Oftentimes the set X will be clear from the context,
and then one simply writes prn instead of prX;n.
*
It is clear that these coordinate functions are continuous. We can now define continuous calculi:
Definition 3.4**.**
*Let A be an ordered ∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH,
then a continuous calculus for a1,…,aN is a triple (X,I,Φ)
of a closed subset X of \mathbbmRN, an intermediate ∗/̄subalgebra I
of C(X) with prn∈I for all n∈{1,…,N}, and
a positive unital ∗/̄homomorphism Φ:I→A fulfilling Φ(prn)=an
for all n∈{1,…,N}.
*
Even though it is not explicitly required in the definition, such a continuous calculus for
an N-tuple of Hermitian elements a1,…,aN can only exist if a1,…,aN are
pairwise commuting, because the coordinate functions pr1,…,prN are pairwise commuting.
Neither existence nor uniqueness of continuous calculi for fixed algebra elements a1,…,aN
are clear. It is especially possible that there are many different continuous calculi on different domains.
One would like to have a most general one:
Definition 3.5**.**
*Let A be an ordered ∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH,
then a continuous calculus (X,I,Φ) for a1,…,aN is called universal
if every continuous calculus (Y,J,Ψ) for a1,…,aN factors through
(X,I,Φ) in the following sense: X⊆Y and for every f∈J, its restriction
f∣X to a function on X is an element of I and Φ(f∣X)=Ψ(f) holds.
*
It is immediately clear that the universal continuous calculus is uniquely determined (if it exists):
Proposition 3.6**.**
*Let A be an ordered ∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH,
and let (X,I,Φ) and (Y,J,Ψ) be two universal continuous calculi
for a1,…,aN, then X=Y, I=J and Φ=Ψ.
*
It thus makes sense to define:
Definition 3.7**.**
*Let A be an ordered ∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH
such that the universal continuous calculus (X,I,Φ) exists.
Then X is called the spectrum of a1,…,aN and is denoted by
spec(a1,…,aN):=X⊆\mathbbmRN. Similarly, define
\mathcal{F}(a_{1},\dots,a_{N})\coloneqq\mathcal{I}\subseteq\mathscr{C}\big{(}\operatorname{\mathrm{spec}}(a_{1},\dots,a_{N})\big{)}
and Γa1,…,aN:=Φ:F(a1,…,aN)→A.
*
If (X,I,Φ) is a continuous calculus for a tuple a1,…,aN, N∈\mathbbmN, of Hermitian elements
of an ordered ∗/̄algebra A, then one can in general not expect I to be finitely generated as a ∗/̄algebra.
However, it will become clear from the construction of continuous calculi that I still fulfils a
weaker finiteness condition that will be discussed in the remainder of this section:
Definition 3.8**.**
*Let A be an ordered ∗/̄algebra and p∈AH+, then p is said to be proper in A
if for every λ∈]0,∞[ and every a∈AH+ there exist μ∈[0,∞[ and b∈AH+
such that for every ϵ∈]0,∞[ one finds k∈\mathbbmN0 for which the estimate a≤μ\mathbbm1+(p/λ)k+ϵb
holds.
*
For example, if A is an Archimedean ordered ∗/̄algebra in which every element is uniformly bounded, i.e. Abd=A,
then every p∈AH+ is proper in A because a≤∥a∥∞\mathbbm1 for all a∈AH+.
Less trivial examples will be discussed in the next Section 4.
Definition 3.9**.**
*Let A be a Su∗/̄algebra and S⊆Abd, then ⟨⟨S⟩⟩C∗ denotes the C∗/̄subalgebra
of Abd that is generated by S, i.e. the closure in Abd with respect to the C∗/̄norm ∥\ignorespaces⋅\ignorespaces∥∞
of the unital ∗/̄subalgebra that is generated by S.
*
As the restriction of ∥\ignorespaces⋅\ignorespaces∥∞ to the uniformly bounded elements is a C∗/̄norm,
⟨⟨S⟩⟩C∗ is indeed a C∗/̄algebra.
Lemma 3.10**.**
*Let A be a commutative Su∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH.
Then one has an(\mathbbm1+a12+⋯+aN2)−1∈Abd for all n∈{1,…,N}.
*
Proof**:**
From \mathbbm1+a12+⋯+aN2≥\mathbbm1 it follows that 0≤(\mathbbm1+a12+⋯+aN2)−1≤\mathbbm1,
and therefore
[TABLE]
shows that an(\mathbbm1+a12+⋯+aN2)−1∈Abd for all n∈{1,…,N}.
\boxempty
Because of this, the next definition makes sense:
Definition 3.11**.**
*A commutative Su∗/̄algebra A is said to be of finite type if there are N∈\mathbbmN and
a1,…,aN∈AH for which p:=\mathbbm1+a12+⋯+aN2
is proper in A and which generate A in the sense that for every b∈A one finds
numerator and denominator bn,bd∈⟨⟨{p−1,a1p−1,…,aNp−1}⟩⟩C∗
such that bd is invertible in A and b=bnbd−1.
In this case a1,…,aN are called generators of A.
*
It should not come as a surprise that the surjective image of a commutative Su∗/̄algebra of finite type is again of finite type:
Proposition 3.12**.**
*Let A and B be two commutative Su∗/̄algebras and Φ:A→B
a surjective positive unital ∗/̄homomorphism. If A is of finite type with generators
a1,…,aN∈AH, N∈\mathbbmN, then B is also of finite type with
generators Φ(a1),…,Φ(aN).
*
Proof**:**
Write p:=\mathbbm1A+a12+⋯+aN2, then Φ(p)=\mathbbm1B+Φ(a1)2+⋯+Φ(aN)2 is proper:
Indeed, for every λ∈]0,∞[ and every element of BH+,
which can be expressed as Φ(a) with a∈AH by taking the real part
of any preimage under Φ, there exist μ∈[0,∞[ and a~∈AH+
such that for every ϵ∈]0,∞[ one finds k∈\mathbbmN0 for which the estimate
21(\mathbbm1A+a2)≤μ\mathbbm1A+(p/λ)k+ϵa~ holds because p is proper in A.
As a consequence, \Phi(a)\leq\frac{1}{2}(\mathbbm{1}_{\mathcal{B}}+\Phi(a)^{2})=\Phi\big{(}\frac{1}{2}(\mathbbm{1}_{\mathcal{A}}+a^{2})\big{)}\leq\mu\mathbbm{1}_{\mathcal{B}}+(\Phi(p)/\lambda)^{k}+\epsilon\Phi(\tilde{a}) holds with Φ(a~)∈BH+.
Now given b∈B, then there is some a~∈A with Φ(a~)=b.
As A is of finite type, there are a~n,a~d∈⟨⟨{p−1,a1p−1,…,aNp−1}⟩⟩C∗
such that a~d is invertible in A and a~=a~na~d−1.
Using that ∥Φ(a)∥∞≤∥a∥∞ for all a∈Abd,
one sees that the image of ⟨⟨{p−1,a1p−1,…,aNp−1}⟩⟩C∗ under Φ
is contained in the C∗/̄subalgebra ⟨⟨{Φ(p)−1,Φ(a1)Φ(p)−1,…,Φ(aN)Φ(p)−1}⟩⟩C∗
of Bbd. This especially applies to Φ(a~n) and Φ(a~d),
and Φ(a~d) is invertible in B with inverse
Φ(a~d)−1=Φ(a~d−1),
and b=Φ(a~)=Φ(a~n)Φ(a~d)−1 holds.
\boxempty
4 Proper Su∗/̄Algebras of Continuous Functions
Let X be a (possibly empty) topological space.
In this section, certain intermediate ∗/̄subalgebras of Su∗/̄algebras of
C(X) are examined.
Recall that a function p∈C(X) is usually said to be a proper function if
the preimage of every compact subset of \mathbbmC under p is again compact. If
p∈C(X)H+, this is equivalent to p^{-1}\big{(}[0,n]\big{)}
being compact for all n∈\mathbbmN. In this case let
U_{n}\coloneqq p^{-1}\big{(}{]{-\infty},n[}\big{)}=p^{-1}\big{(}{[0,n[}\big{)}
and K_{n}\coloneqq p^{-1}\big{(}[0,n]\big{)}, then Un⊆Kn⊆Un+1⊆Kn+1
holds for all n∈\mathbbmN and X=⋃n∈\mathbbmNUn=⋃n∈\mathbbmNKn.
As Un is open and Kn compact for all n∈\mathbbmN, it follows that X is locally
compact and admits an exhaustion by compact sets, i.e. the sequence (Kn)n∈\mathbbmN
covers X and is strictly increasing in the sense that Kn is contained in the
interior of Kn+1 for all n∈\mathbbmN.
Definition 4.1**.**
*Let X be a locally compact Hausdorff space, then a
proper Su∗/̄algebra of continuous functions on X is an intermediate ∗/̄subalgebra
I of C(X) with the property that there exists a proper
function p∈IH+.
*
By Proposition 3.2, such an intermediate ∗/̄subalgebra
of C(X) indeed is a Su∗/̄algebra.
If X is a compact Hausdorff space, then there exists exactly one
proper Su∗/̄algebra of continuous functions on X, namely the commutative C∗-algebra C(X) itself.
Another example is, for every N∈\mathbbmN, the ∗/̄algebra of all continuous functions on \mathbbmRN that are bounded by a polynomial function.
More generally, it is easy to check that continuous calculi are defined on proper Su∗/̄algebras of continuous functions:
Proposition 4.2**.**
*Let (X,I,Φ) be a continuous calculus for some N-tuple
of Hermitian elements of some ordered ∗/̄algebra, then
p:=\mathbbm1+pr12+⋯+prN2∈IH+
is a proper function and I
is a proper Su∗/̄algebra of continuous functions on X.
*
The notions of proper functions and
of proper elements in ordered ∗/̄algebras, Definition 3.8, are closely related:
Proposition 4.3**.**
*Let I be a proper Su∗/̄algebra of continuous functions on some locally
compact Hausdorff space X, then an element q∈IH+ is a proper function
if and only if it is proper in I.
*
Proof**:**
As I is a proper Su∗/̄algebra of continuous functions on X, there
exists a proper function p∈IH+.
First assume that some element q∈IH+ is proper in I.
Then for every λ∈]0,∞[ there are μ∈[0,∞[ and h∈IH+
such that for all ϵ∈]0,∞[ one finds k∈\mathbbmN0 for which the estimate
p≤μ\mathbbm1+(q/λ)k+ϵh holds. Let K be the closed
(possibly empty) preimage K:=q−1([0,λ]), then
p(x)≤μ+qk(x)/λk+ϵh(x)≤μ+1+ϵh(x) for all x∈K independently of k,
hence p(x)≤μ+1 for all x∈K. So K is a closed subset of
the preimage p−1([0,μ+1]), which is compact because p is a proper function.
This shows that K is also compact and one concludes that q is a proper function.
Conversely, assume that some element q∈IH+ is a proper function.
Given λ∈]0,∞[ and f∈IH+, then let K be the
compact (possibly empty) preimage K:=q−1([0,λ+1]) and define
μ as the maximum of the set {f(x)∣x∈K}∪{0} and g:=f2∈IH+.
Then for every ϵ∈]0,∞[ and all x∈X one can check that the estimate f(x)≤ϵ−1+ϵg(x) holds
(consider the cases f(x)≤ϵ−1 and f(x)≥ϵ−1 separately)
and there is a k∈\mathbbmN0 for which ϵ−1≤(λ+1)k/λk.
Thus f(x)≤qk(x)/λk+ϵg(x) holds for all x∈X\K and consequently f≤μ\mathbbm1+(q/λ)k+ϵg,
which shows that q is proper in I.
\boxempty
Because of the above Proposition 4.3 it will no longer be necessarily
to destinguish between “proper functions” and “proper elements” in proper Su∗/̄algebras
of continuous functions. However, note that Proposition 4.3 does
not generalize to arbitrary intermediate ∗/̄algebras of C(X) for arbitrary
locally compact Hausdorff spaces X: For example, the constant 1-function on \mathbbmR
is not a proper function, but is proper in C(\mathbbmR)bd. Nevertheless,
one could extend all functions in C(\mathbbmR)bd to the Stone-Čech compactification
of \mathbbmR, on which \mathbbm1 is a proper function, so the only problem is the “wrong” domain
of definition of functions in C(\mathbbmR)bd.
In a proper Su∗/̄algebra of continuous functions I on a locally compact Hausdorff space X
one can construct the pointwise square root f∈IH+ for all f∈IH+.
Note that 0≤f≤\mathbbm1+f holds pointwise for all f∈IH+
so that indeed f∈I by Proposition 3.2.
It is also easy to check that this pointwise square root coincides
with the general one discussed in [14].
Similarly, one obtains the pointwise absolute value ∣f∣=f∗f∈IH+ for all f∈I
and the pointwise positive and negative parts f+:=(∣f∣+f)/2∈IH+ and f−:=(∣f∣−f)/2∈IH+
for all f∈IH, i.e. f+(x)=max{f(x),0} and f−(x)=max{−f(x),0} for all x∈X.
These constructions will be quite helpful in the following.
The central result for proper Su∗/̄algebras of continuous functions is the following “Nullstellensatz”,
i.e. a characterization of the vanishing ideals:
Proposition 4.4**.**
*Let X be a locally compact Hausdorff space, I a proper Su∗/̄algebra of
continuous functions on X and V a closed ideal of I. Define
the closed subset Z_{\mathcal{V}}\coloneqq\big{\{}\,x\in X\;\big{|}\;\forall_{f\in\mathcal{V}}:f(x)=0\,\big{\}}
of X, which describes the common zeros of the functions in V,
then \mathcal{V}=\big{\{}\,f\in\mathcal{I}\;\big{|}\;\forall_{z\in Z_{\mathcal{V}}}:f(z)=0\,\big{\}}.
*
Proof**:**
As f(z)=0 for all f∈V and all z∈ZV by definition of ZV, the inclusion
“⊆” is clear.
Conversely, consider the case of a function f∈I that fulfils
f(z)=0 for all z∈ZV, and define g^:=(p+\mathbbm1)−1(∣f∣+\mathbbm1)−1f∈C(X)bd,
where p∈IH+ is a proper function. In order to prove that
f∈V, it is sufficient to construct a sequence
(gn)n∈\mathbbmN in V that converges against g^
because V is a closed ideal of I by assumption.
Fix n∈\mathbbmN. The estimate ∣g^(x)∣<1/(1+n) holds for all
x\in X\backslash p^{-1}\big{(}[0,n]\big{)}. Because of this, K_{n}\coloneqq\big{\{}\,x\in X\;\big{|}\;\lvert\hat{g}(x)\rvert\geq 1/(1+n)\,\big{\}}
is a closed subset of the compact preimage p^{-1}\big{(}[0,n]\big{)}, hence is again compact.
Note that Kn∩ZV=∅, because g^ is non-zero in all
points of Kn but zero on ZV. So for all x∈Kn there exists an hx∈V
with hx(x)=0. The open sets \big{\{}\,\tilde{x}\in X\;\big{|}\;h_{x}(\tilde{x})\neq 0\,\big{\}} for all x∈Kn
cover Kn, so there exists M∈\mathbbmN and x1,…,xM∈Kn such that
K_{n}\subseteq\bigcup_{m=1}^{M}\big{\{}\,\tilde{x}\in X\;\big{|}\;h_{x_{m}}(\tilde{x})\neq 0\,\big{\}}. Thus
en:=λ∑m=1Mhxm∗hxm∈V∩IH+
fulfils en(x)>0 for all x∈Kn and all choices of λ∈]0,∞[,
hence even en(x)≥1 for all x∈Kn if λ is chosen sufficiently
large because Kn is compact.
The above construction yields a sequence (en)n∈\mathbbmN of functions in
V∩IH+ with the property that en(x)≥1
for all those x∈X and n∈\mathbbmN for which ∣g^(x)∣≥1/(1+n). By multiplying each
en with a suitable function dn∈C(X)bd⊆I one can now
construct a sequence \mathbbmN∋n↦gn:=endn∈V
that converges against g^: One possible choice is
[TABLE]
then gn(x)=g^(x) holds for all x∈X with ∣g^(x)∣≥1/(1+n),
and at the remaining points one has ∣gn(x)∣≤∣g^(x)∣<1/(1+n), hence
∣gn(x)−g^(x)∣≤∣gn(x)∣+∣g^(x)∣<2/(1+n).
\boxempty
Every locally compact Hausdorff space X is a Tychonoff space [9, Chap. 5, Thms. 17-18],
i.e. X is Hausdorff and for every closed subset Z of X and every x∈X\Z
there exists a continuous function f:X→[0,1]
such that f(x)=1 and f(z)=0 for all z∈Z. This yields:
Corollary 4.5**.**
*Let X be a locally compact Hausdorff space and I a proper Su∗/̄algebra of
continuous functions on X. Assigning to every closed subset Z of X its
vanishing ideal VZ, i.e. the closed ∗/̄ideal
\mathcal{V}_{Z}\coloneqq\big{\{}\,f\in\mathcal{I}\;\big{|}\;\forall_{z\in Z}:f(z)=0\,\big{\}},
yields a bijection between the closed subsets of X and the closed ∗/̄ideals of I.
Its inverse is the assignment of the closed set ZV of common zeros to every closed
∗/̄ideal V of I as in the previous Proposition 4.4.
*
Proof**:**
It is easy to check that the evaluation functionals I∋f↦f(x)∈\mathbbmC
are continuous unital ∗/̄homomorphisms for all x∈X, so the vanishing ideals as the intersection of
the kernels of such morphisms are closed ∗/̄ideals of I.
Moreover, the mapping Z↦VZ is surjective onto the closed ∗/̄ideals of I
by the previous Proposition 4.4, and it is also injective:
If Z is a closed subset of X, then the set of all
common zeros of the functions in VZ is again Z,
because for every x∈X\Z there exists an f∈VZ with f(x)=1
as X is a Tychonoff space. This also shows that the inverse of the construction of vanishing ideals
Z↦VZ is the construction of the set of common zeros V↦ZV.
\boxempty
For unital ∗/̄homomorphisms to an Archimedean ordered ∗/̄algebra one thus finds:
Corollary 4.6**.**
Let X be a locally compact Hausdorff space, I a proper Su∗/̄algebra of
continuous functions on X and Φ:I→A a unital ∗/̄homomorphism
to an Archimedean ordered ∗/̄algebra A. Then Φ automatically is
a positive unital ∗/̄homomorphism and continuous, and there is a unique closed subset Z of X such that
kerΦ=VZ like in the previous Corollary 4.5.
Moreover, for every f∈IH with Φ(f)∈AH+
one has Φ(f)=Φ(f+) with f+∈IH+ the positive part
of f. Similarly, for all ρ∈]0,∞[ and every f∈I fulfilling
∥Φ(f)∥∞≤ρ
one has Φ(γρ∘f)=Φ(f) with the continuous function
γρ:\mathbbmC→\mathbbmC defined as
[TABLE]
*for all z∈\mathbbmC.
*
Proof**:**
Given f∈IH+, then Φ(f)=Φ(f)2∈AH+,
so Φ is a positive unital ∗/̄homomorphism and therefore is continuous. This shows that
its kernel is a closed ∗/̄ideal of I, thus kerΦ=VZ for a
unique closed subset Z of X by the previous Corollary 4.5.
Moreover, assume that some function f∈IH fulfils Φ(f)∈AH+,
then the negative part of f
fulfils 0\leq\Phi\big{(}(f_{-})^{3}\big{)}=-\Phi(f_{-})\Phi(f)\Phi(f_{-})\leq 0
because f−≥0, f−f+=0 and Φ(f)≥0, so
(f−)3∈VZ and consequently also f−∈VZ.
We conclude that Φ(f)=Φ(f+).
Similarly, for ρ∈]0,∞[ and f∈I with ∥Φ(f)∥∞≤ρ one has
\Phi\big{(}\rho^{2}\mathbbm{1}_{\mathcal{I}}-f^{*}f\big{)}=\rho^{2}\mathbbm{1}_{\mathcal{A}}-\Phi(f)^{*}\Phi(f)\geq 0
so that (ρ2\mathbbm1I−f∗f)−∈VZ.
As γρ∘f and f differ only in points x∈X for which ∣f(x)∣>ρ,
this shows that f−(γρ∘f)∈VZ, so Φ(γρ∘f)=Φ(f).
\boxempty
Under the bijection between closed subsets and closed ∗/̄ideals from Corollary 4.5,
the subsets which consist of only one single point correspond to the closed ∗/̄ideals with codimension 1.
Thus one finds that unital ∗/̄homomorphisms between proper Su∗/̄algebras of continuous
functions are always of particularly simple form:
Proposition 4.7**.**
*Let X and Y be two locally compact Hausdorff spaces and let I and J
be two proper Su∗/̄algebras of continuous functions on X and Y, respectively. Then for every
unital ∗/̄homomorphism Φ:I→J there exists a unique
continuous map ϕ:Y→X such that Φ(f)=f∘ϕ holds for all f∈I.
*
Proof**:**
Let such a unital ∗/̄homomorphism Φ:I→J be given.
For every y∈Y, define the unital ∗/̄homomorphism Φy:I→\mathbbmC,
f↦Φy(f):=Φ(f)(y). Then, as discussed in the previous Corollary 4.6,
there exists a unique closed subset Zy of X such that kerΦy=VZy.
Moreover, Zy cannot be empty because then kerΦy=VZy=I would contradict
Φy(\mathbbm1I)=Φ(\mathbbm1I)(y)=\mathbbm1J(y)=1. Now given any
z∈Zy, then on the one hand kerΦy=VZy⊆V{z},
and on the other hand one finds for all f∈V{z} that
f−Φy(f)\mathbbm1I∈kerΦy⊆V{z}
implies that f(z)−Φy(f)=0 holds, so Φy(f)=f(z)=0 and thus f∈kerΦy.
It follows that V{z}=kerΦy=VZy
and therefore {z}=Zy by uniqueness of Zy, so Zy consists of exactly one single point of X.
It is now possible to define a map ϕ:Y→X, y↦ϕ(y),
with ϕ(y) being the unique point for which {ϕ(y)}=Zy. Then
\Phi(f)(y)=\Phi_{y}(f)=\Phi_{y}\big{(}f-f(\phi(y))\mathbbm{1}_{\mathcal{I}}\big{)}+f(\phi(y))=f(\phi(y))
holds for all f∈I and all y∈Y because f−f(ϕ(y))\mathbbm1I∈V{ϕ(y)}=kerΦy,
so Φ(f)=f∘ϕ for all f∈I. Moreover, ϕ is continuous: Indeed,
the bijective correspondence between closed subsets of X and closed ∗/̄ideals of I
from Corollary 4.5 allows to describe the preimage ϕ−1(Z) of some closed subset Z of X as
[TABLE]
which is a closed subset of Y because Φ(f) is a continuous function on Y for all f∈I.
Finally, it only remains to check that ϕ is uniquely determined: So let ϕ:Y→X
be any map for which Φ(f)=f∘ϕ holds and let y∈Y be given, then for all f∈kerΦy=VZy
one has 0=Φ(f)(y)=f(ϕ(y)), i.e. ϕ(y) is a common zero of all f∈VZy and therefore
ϕ(y)∈Zy by Corollary 4.5. As Zy
consists of exactly one single element of X this means that {ϕ(y)}=Zy.
\boxempty
Note that a statement analogous to the above Proposition 4.7
is not true if I=C(X)bd for non-compact X and if Y={∗} is the topological
space with only one single point: In this case C({∗})≅\mathbbmC and
the set of unital ∗/̄homomorphisms from the commutative C∗/̄algebra C(X)bd
to \mathbbmC is weak/̄∗-compact as a consequence of the Banach-Alaoglu theorem
and especially does not consist of only the evaluation functionals C(X)bd∋f↦f(x)∈\mathbbmC≅C({∗})
at points x∈X. This means that there are unital ∗/̄homomorphisms
C(X)bd→C({∗}) that are not of the form
f↦f∘ι with ι:{∗}→X, ι(∗)=x.
So the assumption that IH+
contains a proper function is crucial.
5 Universal Continuous Calculi – Construction
While proper Su∗/̄algebras of continuous functions automatically have very well-behaved order properties,
see e.g. Corollary 4.6, this is not true for general ordered ∗/̄algebras,
especially not for \mathbbmC[t1,…,tN] with the pointwise order. In order to extend the polynomial calculus to a continuous
calculus it will therefore be necessary to show that similar properties are indeed fulfilled in special cases,
which requires some variant of the Positivstellensatz.
The universal continuous calculus for N∈\mathbbmN pairwise commuting Hermitian elements a1,…,aN of a Su∗/̄algebra A can be constructed
in four steps: First one constructs a positive unital ∗/̄homomorphism Φrt from a ∗/̄algebra Crt(\mathbbmRN) of certain rational
functions to A. Its restriction to some bounded rational functions can then be extended continuously to a (necessarily positive) unital ∗/̄homomorphism
Φ\SS defined on C\SS(\mathbbmRN), the continuous functions on \mathbbmRN that extend continuously to the one-point compactification \mathbbmRN∪{∞}≅\SSN.
In a third step, the domain of definition of Φ\SS is extended to a proper Su∗/̄algebra I\mathbbmR of continuous functions on \mathbbmRN,
which is essentially the largest possible one that admits an extension Φ\mathbbmR:I\mathbbmR→A of Φ\SS.
The final step is to divide out the kernel of Φ\mathbbmR and to interpret the resulting quotient I\mathbbmR/kerΦ\mathbbmR
as a proper Su∗/̄algebra of continuous functions on a closed subset spec(a1,…,aN) of \mathbbmRN. It will also become clear from the
construction that the universal continuous calculus for a1,…,aN maps to the bicommutant {a1,…,aN}′′.
Lemma 5.1**.**
*If π∈\mathbbmC[s,t1,…,tN]H with N∈\mathbbmN fulfils
\pi\big{(}(1+x_{1}^{2}+\dots+x_{N}^{2})^{-1},x_{1},\dots,x_{N}\big{)}\geq 0 for all x∈\mathbbmRN, then
\pi\big{(}(\mathbbm{1}+a_{1}^{2}+\dots+a_{N}^{2})^{-1},a_{1},\dots,a_{N}\big{)}\in\mathcal{A}^{+}_{\textup{H}} holds for every
Archimedean ordered ∗/̄algebra A and every pairwise commuting N-tuple a1,…,aN∈AH
for which \mathbbm1+a12+⋯+aN2 is invertible.
*
Proof**:**
Let π∈\mathbbmC[s,t1,…,tN]H be such a polynomial fulfilling
\pi\big{(}(1+x_{1}^{2}+\dots+x_{N}^{2})^{-1},x_{1},\dots,x_{N}\big{)}\geq 0 for all x∈\mathbbmRN.
There are K,L∈\mathbbmN0 for which π can be expanded as
[TABLE]
with coefficients πk,ℓ∈\mathbbmR. Using these coefficients, one can now construct a homogeneous polynomial
ρ∈\mathbbmC[u1,…,uN,v1,…,vN,w]H of degree 2K+L as
[TABLE]
Then z^{-(2K+L)}\rho(x,y,z)=\rho(x/z,y/z,1)=\lambda^{K}\pi\big{(}\lambda^{-1},(x_{1}-y_{1})/z_{1},\dots,(x_{N}-y_{N})/z_{N}\big{)}\geq 0
with λ:=1+(x1−y1)2/z12+⋯+(xN−yN)2/zN2
holds for all (x,y,z)∈\mathbbmRN×\mathbbmRN×(\mathbbmR\{0}), which
shows that ρ(x,y,z)≥0 for all (x,y,z)∈\mathbbmRN×\mathbbmRN×]0,∞[,
and thus even for all (x,y,z)∈\mathbbmRN×\mathbbmRN×[0,∞[ by continuity.
Fix ϵ∈]0,∞[, then the homogeneous polynomial ρ+ϵ(u1+⋯+uN+v1+⋯+vN+w)2K+L
is strictly positive on ([0,∞[N×[0,∞[N×[0,∞[)\{(0,0,0)},
so by a theorem of Pólya, [7, Thm. 56],
there exists an M∈\mathbbmN0 such that
[TABLE]
can be expanded as
[TABLE]
with positive coefficients σϵ;e,f∈[0,∞[.
Now let A be any Archimedean ordered ∗/̄algebra, a1,…,aN∈AH
pairwise commuting and such that p:=\mathbbm1+a12+⋯+aN2 is invertible in A.
Note that p and hence also p−1 commute with all a1,…,aN.
Define the element
[TABLE]
of A, where χ∈]0,∞[ will be specified later on. Then bϵ
is a sum of squares of Hermitian elements of A, so bϵ∈AH+.
In order to simplify the expression for bϵ, note that
(χ\mathbbm1+χ−1an)2/4−(χ\mathbbm1−χ−1an)2/4=an, and similarly
(χ\mathbbm1+χ−1an)2/4+(χ\mathbbm1−χ−1an)2/4=(χ4\mathbbm1+an2)/(2χ2)
holds for all n∈{1,…,N}, so
[TABLE]
One can choose χ∈]0,∞[ in such a way that Nχ4+2χ2=1, namely χ=((N−2+N−1)1/2−N−1)1/2,
and with this choice one now finds that
[TABLE]
As bϵ by construction commutes with all an, n∈{1,…,N}, so that bϵan2=anbϵan≥0,
it follows that
[TABLE]
But as this is true for all ϵ∈]0,∞[ and as A is Archimedean,
one has pKπ(p−1,a1,…,aN)≥0, and thus, by a similar argument,
[TABLE]
because pKπ(p−1,a1,…,aN) commutes with all a1,…,aN.
\boxempty
One can now construct a “rational calculus”:
Definition 5.2**.**
*For N∈\mathbbmN let Crt(\mathbbmRN) be the unital ∗/̄subalgebra of C(\mathbbmRN)
that is generated by the functions pr1,…,prN and (\mathbbm1+pr12+⋯+prN2)−1,
endowed with the order on Hermitian elements that it inherits from C(\mathbbmRN), i.e. the pointwise one.
*
Proposition 5.3**.**
*Let A be an Archimedean ordered ∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH
pairwise commuting, and assume that \mathbbm1+a12+⋯+aN2 has a multiplicative inverse (\mathbbm1+a12+⋯+aN2)−1∈A.
Then there exists a unique positive unital ∗/̄homomorphism
Φrt:Crt(\mathbbmRN)→A fulfilling Φrt(prn)=an
for all n∈{1,…,N}. Moreover, Φrt(r)∈{a1,…,aN}′′ for all r∈Crt(\mathbbmRN).
*
Proof**:**
Let [\ignorespaces⋅\ignorespaces]rt:\mathbbmC[s,t1,…,tN]→Crt(\mathbbmRN), π↦[π]rt be the unital ∗/̄homomorphism
that maps s to (\mathbbm1Crt(\mathbbmRN)+pr12+⋯+prN2)−1 and tn to prn for all n∈{1,…,N},
then [\ignorespaces⋅\ignorespaces]rt maps surjectively onto Crt(\mathbbmRN)
by definition of Crt(\mathbbmRN).
If
a polynomial π∈\mathbbmC[s,t1,…,tN]H fulfils [π]rt∈Crt(\mathbbmRN)H+,
then [π]rt is pointwise positive on \mathbbmRN by definition of the order on Crt(\mathbbmRN)H,
so \pi\big{(}(1+x_{1}^{2}+\dots+x_{N}^{2})^{-1},x_{1},\dots,x_{N}\big{)}=[\pi]_{\mathrm{rt}}(x_{1},\dots,x_{N})\geq 0 for all x∈\mathbbmRN,
and Lemma 5.1 shows that \pi\big{(}(\mathbbm{1}_{A}+a_{1}^{2}+\dots+a_{N}^{2})^{-1},a_{1},\dots,a_{N}\big{)}\in\mathcal{A}_{\textup{H}}^{+}.
Especially if [π]rt=0, then \pi\big{(}(\mathbbm{1}_{A}+a_{1}^{2}+\dots+a_{N}^{2})^{-1},a_{1},\dots,a_{N}\big{)}\in\mathcal{A}_{\textup{H}}^{+}\cap(-\mathcal{A}_{\textup{H}}^{+})=\{0\}.
More generally, if [π]rt=0 for any (not necessarily Hermitian) π∈\mathbbmC[s,t1,…,tN], then
it follows from
[Re(π)]rt=0=[Im(π)]rt that
\pi\big{(}(\mathbbm{1}_{A}+a_{1}^{2}+\dots+a_{N}^{2})^{-1},a_{1},\dots,a_{N}\big{)}=0. All of this together shows that the map
Φrt:Crt(\mathbbmRN)→A,
[TABLE]
is a well-defined positive unital ∗/̄homomorphism and one has \Phi_{\mathrm{rt}}(\mathrm{pr}_{n})=\Phi_{\mathrm{rt}}\big{(}[t_{n}]_{\mathrm{rt}}\big{)}=a_{n}
for all n∈{1,…,N} and
\Phi_{\mathrm{rt}}\big{(}(\mathbbm{1}_{\mathscr{C}_{\mathrm{rt}}(\mathbbm{R}^{N})}+\mathrm{pr}_{1}^{2}+\dots+\mathrm{pr}_{N}^{2})^{-1}\big{)}=\Phi_{\mathrm{rt}}\big{(}[s]_{\mathrm{rt}}\big{)}=(\mathbbm{1}_{A}+a_{1}^{2}+\dots+a_{N}^{2})^{-1}.
As Crt(\mathbbmRN) is generated as a unital ∗/̄algebra by pr1,…,prN and (\mathbbm1Crt(\mathbbmRN)+pr12+⋯+prN2)−1,
a unital ∗/̄homomorphism Φrt:Crt(\mathbbmRN)→A
is uniquely determined by fixing Φrt(pr1),…,Φrt(prN) because the identity
\Phi_{\mathrm{rt}}\big{(}(\mathbbm{1}_{\mathscr{C}_{\mathrm{rt}}(\mathbbm{R}^{N})}+\mathrm{pr}_{1}^{2}+\dots+\mathrm{pr}_{N}^{2})^{-1}\big{)}=\big{(}\mathbbm{1}_{A}+\Phi_{\mathrm{rt}}(\mathrm{pr}_{1})^{2}+\dots+\Phi_{\mathrm{rt}}(\mathrm{pr}_{N})^{2}\big{)}{}^{-1} necessarily holds. Moreover, as (\mathbbm1A+a12+⋯+aN2)−1∈{a1,…,aN}′′ and an∈{a1,…,aN}′′ for all n∈{1,…,N}
it also follows that Φrt(r)∈{a1,…,aN}′′ for all r∈Crt(\mathbbmRN).
\boxempty
This “rational calculus” can now be extended to continuous uniformly bounded functions whose limit at ∞ exists:
Definition 5.4**.**
For N∈\mathbbmN let \SSN be the one-point compactification of \mathbbmRN, i.e. \SSN:=\mathbbmRN∪{∞} as a set
and a subset U of \SSN is open if and only if either U is an open subset of \mathbbmRN or if
\SSN\U is a compact subset of \mathbbmRN. Moreover, define
[TABLE]
*where \ignorespaces⋅\ignorespaces∣\mathbbmRN denotes restriction of a function defined on \SSN to \mathbbmRN.
*
Some important properties of C\SS(\mathbbmRN) are:
Lemma 5.5**.**
*Given N∈\mathbbmN, then C\SS(\mathbbmRN) is a closed unital ∗/̄subalgebra of C(\mathbbmRN)bd.
If some function f∈C(\mathbbmRN) vanishes at ∞, i.e. if it has the property that for every ϵ∈]0,∞[
there is a compact subset K of \mathbbmRN such that ∣f(x)∣<ϵ holds for all x∈\mathbbmRN\K,
then f∈C\SS(\mathbbmRN); especially r0:=(\mathbbm1+pr12+⋯+prN2)−1∈C\SS(\mathbbmRN)
and rn:=prn(\mathbbm1+pr12+⋯+prN2)−1∈C\SS(\mathbbmRN) for all n∈{1,…,N}.
Moreover, C\SS(\mathbbmRN) is the C∗/̄subalgebra of C(\mathbbmRN)bd that is generated by {r0,r1,…,rN}
and the unital ∗/̄subalgebra C\SS(\mathbbmRN)∩Crt(\mathbbmRN) of C\SS(\mathbbmRN)
is dense in C\SS(\mathbbmRN).
*
Proof**:**
As the restriction map \ignorespaces⋅\ignorespaces∣\mathbbmRN:C(\SSN)→C(\mathbbmRN) is a positive unital ∗/̄homomorphism,
its image C\SS(\mathbbmRN) is a unital ∗/̄subalgebra of C(\mathbbmRN). As C(\SSN)bd=C(\SSN) due
to the compactness of \SSN, it even follows that C\SS(\mathbbmRN)⊆C(\mathbbmRN)bd by positivity of \ignorespaces⋅\ignorespaces∣\mathbbmRN.
Moreover, this positive unital ∗/̄homomorphism \ignorespaces⋅\ignorespaces∣\mathbbmRN is injective and even an order embedding because
\mathbbmRN is dense in \SSN. This implies that the restriction map is isometric, i.e. ∥f∣\mathbbmRN∥∞=∥f∥∞
for all f∈C(\SSN), and from completeness of C(\SSN) it now follows that C\SS(\mathbbmRN)
is closed in C(\mathbbmRN)bd.
If some function f∈C(\mathbbmRN)bd has the property that for every ϵ∈]0,∞[
there is a compact subset K of \mathbbmRN such that ∣f(x)∣≤ϵ holds for all x∈\mathbbmRN\K,
then the function fext:\SSN→\mathbbmC that is defined as fext(x):=f(x) for all
x∈\mathbbmRN and fext(∞):=0 is continuous, and so f=fext∣\mathbbmRN∈C\SS(\mathbbmRN).
It is easy to check that this condition is fulfilled by the functions rn with n∈{0,…,N} so that
one can construct such extensions rnext∈C(\SSN) for which rn=rnext∣\mathbbmRN∈C\SS(\mathbbmRN).
Finally, note that these extended functions rnext form a point-separating set of functions on \SSN,
i.e. for all x,y∈\SSN with x=y there is an n∈{0,1,…,N} such that rnext(x)=rnext(y):
Indeed, if both x and y are elements of \mathbbmRN but x=y, then there is an n∈{1,…,N} such that prn(x)=prn(y)
and therefore rnext(x)=rnext(y); if exactly one of x and y equals ∞, say x=∞ and y∈\mathbbmRN,
then r0ext(x)=0=r0(y)=r0ext(y). Thus by the Stone-Weierstraß theorem, the unital ∗/̄subalgebra
of C(\SSN) that is generated by these extended functions {r0ext,r1ext,…,rNext} is
dense in C(\SSN). From the continuity of the restriction map \ignorespaces⋅\ignorespaces∣\mathbbmRN it follows
that the unital ∗/̄subalgebra of C\SS(\mathbbmRN) that is generated by the functions {r0,r1,…,rN} is dense in C\SS(\mathbbmRN),
so ⟨⟨{r0,r1,…,rN}⟩⟩C∗=C\SS(\mathbbmRN) because C\SS(\mathbbmRN) is also closed in C(\mathbbmRN)bd.
Moreover, as rn∈C\SS(\mathbbmRN)∩Crt(\mathbbmRN) for all n∈{0,1,…,N}, this also means that
the unital ∗/̄subalgebra C\SS(\mathbbmRN)∩Crt(\mathbbmRN) of C\SS(\mathbbmRN)
is dense in C\SS(\mathbbmRN).
\boxempty
Recall that every positive unital ∗/̄homomorphism between Archimedean ordered ∗/̄algebras is automatically continuous, and that Su∗/̄algebras
by definition are complete and contain a multiplicative inverse for every element of the form \mathbbm1+a12+⋯+aN2 with Hermitian
elements a1,…,aN and N∈\mathbbmN. The above Lemma 5.5 therefore shows that on Su∗/̄algebras, one can continuously extend
the rational calculus from Proposition 5.3 to C\SS(\mathbbmRN):
Definition 5.6**.**
*Let A be a Su∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH pairwise commuting, then define
Φ\SS:C\SS(\mathbbmRN)→A as the continuous extension of the restriction of the positive unital ∗/̄homomorphism
Φrt:Crt(\mathbbmRN)→A from Proposition 5.3
to C\SS(\mathbbmRN)∩Crt(\mathbbmRN).
*
Proposition 5.7**.**
*Let A be a Su∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH pairwise commuting, then the map
Φ\SS:C\SS(\mathbbmRN)→A from the previous Definition 5.6
is a positive unital ∗/̄homomorphism and Φ\SS(f)∈{a1,…,aN}′′ for all f∈C\SS(\mathbbmRN).
The image of C\SS(\mathbbmRN) under Φ\SS is the C∗/̄subalgebra of
Abd that is generated by {p−1,a1p−1,…,aNp−1},
where p:=\mathbbm1A+a12+⋯+aN2∈A.
*
Proof**:**
Φ\SS* is defined as the ∥\ignorespaces⋅\ignorespaces∥∞-continuous extension of a ∥\ignorespaces⋅\ignorespaces∥∞-continuous
unital ∗/̄homomorphism, so continuity of multiplication and ∗/̄involution with respect to the norm ∥\ignorespaces⋅\ignorespaces∥∞
guarantee that Φ\SS is again a unital ∗/̄homomorphism. It is automatically positive: If f∣\mathbbmRN is pointwise
positive for some f∈C(\SSN), then f is also pointwise positive because \mathbbmRN is dense in \SSN.
Because of this, the pointwise square root f∈C(\SSN)H+ is well-defined and Φ\SS(f∣\mathbbmRN)=Φ\SS(f∣\mathbbmRN)2≥0.*
Moreover, as the image of Φrt is a subset of the bicommutant {a1,…,aN}′′ by Proposition 5.3
and as {a1,…,aN}′′ is closed in
A by [14, Prop. 5], the image of Φ\SS also lies in {a1,…,aN}′′.
Finally, the image of C\SS(\mathbbmRN) under Φ\SS is ⟨⟨{p−1,a1p−1,…,aNp−1}⟩⟩C∗:
On the one hand, C\SS(\mathbbmRN) is the C∗/̄subalgebra of C(\mathbbmRN)bd that is generated by
r0=(\mathbbm1C(\mathbbmRN)+pr12+⋯+prN2)−1 and all rn=prnr0 with n∈{1,…,N}
by Lemma 5.5, so its image under the ∥\ignorespaces⋅\ignorespaces∥∞-continuous map Φ\SS is
a subset of ⟨⟨{p−1,a1p−1,…,aNp−1}⟩⟩C∗. On the other hand, let
b∈⟨⟨{p−1,a1p−1,…,aNp−1}⟩⟩C∗ be given. Then there exists a convergent sequence
(ck)k∈\mathbbmN in the unital ∗/̄subalgebra of Abd that is generated by {p−1,a1p−1,…,aNp−1}
with limit limk→∞ck=b. One can even arrange that ∥b−ck∥∞≤2−(k+1) for all k∈\mathbbmN,
hence ∥ck+1−ck∥∞≤2−k. This allows to construct a Cauchy-sequence (gk)k∈\mathbbmN in C\SS(\mathbbmRN)
fulfilling Φ\SS(gk)=ck for all k∈\mathbbmN as follows: Let g1∈C\SS(\mathbbmRN) be any preimage of c1 under
Φ\SS, which exists by construction of the sequence (ck)k∈\mathbbmN. If g1,…,gK have been defined for some K∈\mathbbmN,
let gK+1:=gK+(γ2−K∘δK)∣\mathbbmRN with δK∈C(\SSN) a preimage of
cK+1−cK under Φ\SS∘\ignorespaces⋅\ignorespaces∣\mathbbmRN:C(\SSN)→A and γ2−K:\mathbbmC→\mathbbmC
like in Corollary 4.6. Then \Phi_{\SS}\big{(}g_{K}+(\gamma_{2^{-K}}\circ\delta_{K})|_{\mathbbm{R}^{N}}\big{)}=c_{K}+\Phi_{\SS}(\delta_{K}|_{\mathbbm{R}^{N}})=c_{K+1}
by Corollary 4.6 and because ∥Φ\SS(δK∣\mathbbmRN)∥∞=∥cK+1−cK∥∞≤2−K,
and one also finds that the estimate
∥gK+1−gK∥∞=∥(γ2−K∘δK)∣\mathbbmRN∥∞≤2−K holds because ∣γ2−K(z)∣≤2−K
for all z∈\mathbbmC.
The resulting sequence (gk)k∈\mathbbmN therefore is indeed a Cauchy sequence and its limit f:=limk→∞gk∈C\SS(\mathbbmRN)
fulfils Φ\SS(f)=limk→∞Φ\SS(gk)=limk→∞ck=b.
\boxempty
However, Φ\SS does not describe a continuous calculus simply because prn∈/C\SS(\mathbbmRN)
for n∈{1,…,N}. This will be fixed in the next step:
Proposition 5.8**.**
Let A be a Su∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH pairwise commuting and write
p:=\mathbbm1+pr12+⋯+prN2∈C(\mathbbmRN). Construct C\SS(\mathbbmRN) and Φ\SS:C\SS(\mathbbmRN)→A
like in Definitions 5.4 and 5.6.
Then for every f∈C(\mathbbmRN) the function f∗f+p has a pointwise inverse and
(f∗f+p)−1∈C\SS(\mathbbmRN) and f(f∗f+p)−1∈C\SS(\mathbbmRN) hold. Moreover:
i.)
The set
[TABLE]
has the following property: If for some f∈C(\mathbbmRN) there exists an h∈C\SS(\mathbbmRN)H+ such that
Φ\SS(h) has a multiplicative inverse Φ\SS(h)−1∈A and such that (f∗f+p)−1≥h holds,
then f∈I\mathbbmR. This especially shows that I\mathbbmR is an intermediate ∗/̄subalgebra of
C(\mathbbmRN) and Crt(\mathbbmRN)⊆I\mathbbmR. Moreover, I\mathbbmR
is a commutative Su∗/̄algebra of finite type with generators pr1,…,prN.
2. ii.)
The map Φ\mathbbmR:I\mathbbmR→A,
[TABLE]
has the following property: If for some f∈I\mathbbmR there exists an h∈C\SS(\mathbbmRN) such that
Φ\SS(h) has a multiplicative inverse Φ\SS(h)−1∈A and such that fh∈C\SS(\mathbbmRN) is fulfilled,
then the identity Φ\mathbbmR(f)=Φ\SS(fh)Φ\SS(h)−1 holds. This especially shows that Φ\mathbbmR is a unital
∗/̄homomorphism that extends Φ\SS and Φrt, i.e. Φ\mathbbmR(f)=Φ\SS(f)
for all f∈C\SS(\mathbbmRN) and Φ\mathbbmR(r)=Φrt(r) for all r∈Crt(\mathbbmRN).
Moreover, one has Φ\mathbbmR(f)∈{a1,…,aN}′′ for all f∈I\mathbbmR.
3. iii.)
The triple \big{(}\mathbbm{R}^{N},\mathcal{I}_{\mathbbm{R}},\Phi_{\mathbbm{R}}\big{)} is a continuous calculus for a1,…,aN.
Proof**:**
Given f∈C(\mathbbmRN), then f∗f+p≥\mathbbm1, so f∗f+p is invertible in C(\mathbbmRN).
For every ϵ∈]0,∞[, the preimage Kϵ:=p−1([0,ϵ−1])
is a compact subset of \mathbbmRN (i.e. p is proper) and one finds that
(f∗f+p)−1∣\mathbbmRN\Kϵ<ϵ\mathbbm1 and
∣f∣(f∗f+p)−1∣\mathbbmRN\Kϵ<ϵ\mathbbm1
(for the second estimate, consider the two cases ∣f(x)∣≥ϵ−1/2 and 0≤∣f(x)∣≤ϵ−1/2 separately).
So (f∗f+p)−1,f(f∗f+p)−1∈C\SS(\mathbbmRN) by Lemma 5.5.
Recall also that Φ\SS:C\SS(\mathbbmRN)→A is a positive unital ∗/̄homomorphism
by the previous Proposition 5.7.
For part i.) consider f∈C(\mathbbmRN) and h∈C\SS(\mathbbmRN)H+ with the properties that Φ\SS(h)
has a multiplicative inverse Φ\SS(h)−1∈A and (f∗f+p)−1≥h. Then
\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}h\big{)}\geq\Phi_{\SS}(h)^{2} because (f∗f+p)−1h≥h2,
so \Phi_{\SS}(h)^{-1}\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}h\big{)}\Phi_{\SS}(h)^{-1} is coercive
and consequently is invertible in A because A is symmetric by assumption.
It follows that \Phi_{\SS}\big{(}(f^{*}f+p)^{-1}h\big{)} also is invertible in A, thus also
\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}\big{)} with inverse
\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}\big{)}{}^{-1}=\Phi_{\SS}(h)\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}h\big{)}{}^{-1}=\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}h\big{)}{}^{-1}\Phi_{\SS}(h).
This shows that f∈I\mathbbmR.
For example, as Φ\SS(p−1)=(\mathbbm1+a12+⋯+aN2)−1 by construction of Φ\SS,
which is invertible in A, one can choose h:=ϵp−1 with ϵ∈]0,∞[.
So C(\mathbbmRN)bd⊆I\mathbbmR
because (f∗f+p)−1≥(1+∥f∥∞2)−1p−1 holds for all f∈C(\mathbbmRN)bd,
and also prn∈I\mathbbmR for all n∈{1,…,N} because (prn2+p)−1≥21p−1.
Similarly, given f,g∈I\mathbbmR, then one can choose h:=21(f∗f+p)−1(g∗g+p)−1∈C\SS(\mathbbmRN)H+,
for which \Phi_{\SS}(h)=\frac{1}{2}\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}\big{)}\Phi_{\SS}\big{(}(g^{*}g+p)^{-1}\big{)} indeed is invertible in A. The estimates
[TABLE]
then show that \big{(}(f+g)^{*}(f+g)+p\big{)}{}^{-1}\geq h and \big{(}(fg)^{*}(fg)+p\big{)}{}^{-1}\geq h, so f+g,fg∈I\mathbbmR.
As I\mathbbmR clearly is stable under the ∗/̄involution of C(\mathbbmRN), i.e. pointwise complex conjugation,
we conclude that I\mathbbmR is an intermediate ∗/̄subalgebra of
C(\mathbbmRN) fulfilling prn∈I\mathbbmR for all n∈{1,…,N} and p−1∈C(\mathbbmRN)bd⊆I\mathbbmR,
thus Crt(\mathbbmRN)⊆I\mathbbmR because Crt(\mathbbmRN) is
generated by pr1,…,prN and p−1.
Moreover, I\mathbbmR is a proper Su∗/̄algebra of continuous functions on \mathbbmRN because p is a proper function.
Proposition 4.3 thus shows that p is also a proper element of I\mathbbmR in the sense of
Definition 3.8. One even finds that I\mathbbmR is a commutative Su∗/̄algebra of finite type with generators pr1,…,prN:
Indeed, given f∈I\mathbbmR, then by Lemma 5.5, the two functions fn:=f(f∗f+p)−1∈C\SS(\mathbbmRN) and
fd:=(f∗f+p)−1∈C\SS(\mathbbmRN) are elements of the C∗/̄subalgebra of (I\mathbbmR)bd=C(\mathbbmRN)bd
that is generated by {p−1,pr1p−1,…,prNp−1}.
It is also clear that fd is invertible in I\mathbbmR and f=fnfd−1.
For part ii.) consider the map Φ\mathbbmR:I\mathbbmR→A.
Given f∈C\SS(\mathbbmRN), then one has
[TABLE]
and so Φ\mathbbmR extends Φ\SS. Given functions f∈I\mathbbmR and h∈C\SS(\mathbbmRN)
such that Φ\SS(h) has a multiplicative inverse Φ\SS(h)−1∈A and such that fh∈C\SS(\mathbbmRN) holds,
then
[TABLE]
For example, p−1∈C\SS(\mathbbmRN) and prnp−1∈C\SS(\mathbbmRN) for all n∈{1,…,N}
by Lemma 5.5, and Φ\SS(p−1)=Φrt(p−1)=(\mathbbm1+a12+⋯+aN2)−1
is invertible in A. So
[TABLE]
Moreover, given f,g∈I\mathbbmR, then f(f∗f+p)−1,(f∗f+p)−1,g(g∗g+p)−1,(g∗g+p)−1∈C\SS(\mathbbmRN)
has already been shown, thus fh,gh,(f+g)h,fgh∈C\SS(\mathbbmRN) with h:=(f∗f+p)−1(g∗g+p)−1∈C\SS(\mathbbmRN).
As \Phi_{\SS}(h)=\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}\big{)}\Phi_{\SS}\big{(}(g^{*}g+p)^{-1}\big{)} is invertible in A, one finds that
[TABLE]
and similarly also
[TABLE]
hold. Compatibility of Φ\mathbbmR with the ∗/̄involution follows immediately from the compatibility of Φ\SS with the ∗/̄involution,
so it has been shown that Φ\mathbbmR is a unital ∗/̄homomorphism that extends Φ\SS, especially Φ\mathbbmR(p−1)=Φ\SS(p−1)=Φrt(p−1),
and which fulfils Φ\mathbbmR(prn)=Φrt(prn) for all n∈{1,…,N}. As Crt(\mathbbmRN) is generated
by p−1 and pr1,…,prN, it follows that Φ\mathbbmR also extends Φrt.
Note also that Φ\mathbbmR(f)∈{a1,…,aN}′′ for all f∈I\mathbbmR because \Phi_{\SS}\big{(}f(f^{*}f+p)^{-1}\big{)},\Phi_{\SS}\big{(}(f^{*}f+p)^{-1}\big{)}\in\{a_{1},\dots,a_{N}\}^{\prime\prime}
by Proposition 5.7.
Finally for part iii.) it only remains to combine the previous results: By part i.),
I\mathbbmR is an intermediate ∗/̄subalgebra of C(\mathbbmRN) fulfilling prn∈I\mathbbmR for all n∈{1,…,N}.
By part ii.), Φ\mathbbmR:I\mathbbmR→A is a unital ∗/̄homomorphism and Φ\mathbbmR(prn)=Φrt(prn)=an
for all n∈{1,…,N}. So \big{(}\mathbbm{R}^{N},\mathcal{I}_{\mathbbm{R}},\Phi_{\mathbbm{R}}\big{)} is a continuous calculus for a1,…,aN.
\boxempty
The continuous calculus that has just been constructed is maximal in the following sense:
Corollary 5.9**.**
*Let A be a Su∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH pairwise commuting, and
let \big{(}\mathbbm{R}^{N},\mathcal{I}_{\mathbbm{R}},\Phi_{\mathbbm{R}}\big{)} be the continuous calculus for a1,…,aN that was constructed
in the previous Proposition 5.8. Moreover, let (\mathbbmRN,J,Ψ) be
another continuous calculus for a1,…,aN that is also defined on a space J of continuous functions
on \mathbbmRN. Then J⊆I\mathbbmR and Ψ(f)=Φ\mathbbmR(f) for all f∈J.
*
Proof**:**
As Ψ(prn)=an=Φ\mathbbmR(prn) for all n∈\mathbbmN, and as Crt(\mathbbmRN)
is by definition generated by p−1,pr1,…,prN∈I\mathbbmR∩J
with p:=\mathbbm1+pr12+⋯+prN2∈I\mathbbmR∩J,
it is clear that Crt(\mathbbmRN)⊆I\mathbbmR∩J and that Ψ and Φ\mathbbmR coincide
on Crt(\mathbbmRN). As I\mathbbmR and J are proper Su∗/̄algebras of continuous functions on \mathbbmRN,
the unital ∗/̄homomorphisms Φ\mathbbmR:I\mathbbmR→A and Ψ:J→A
are automatically positive by Corollary 4.6, therefore are continuous and thus coincide even on the closure
of Crt(\mathbbmRN) in I\mathbbmR∩J, which, by Lemma 5.5,
contains C\SS(\mathbbmRN).
Now let any f∈J be given, then f(f∗f+p)−1∈C\SS(\mathbbmRN) and (f∗f+p)−1∈C\SS(\mathbbmRN)
by the previous Proposition 5.8, and
\Phi_{\mathbbm{R}}\big{(}(f^{*}f+p)^{-1}\big{)}=\Psi\big{(}(f^{*}f+p)^{-1}\big{)}=\Psi(f^{*}f+p)^{-1} is invertible in A, so f∈I\mathbbmR
because Φ\mathbbmR extends Φ\SS by the previous Proposition 5.8.
Similarly,
\Psi(f)=\Psi\big{(}f(f^{*}f+p)^{-1}\big{)}\Psi\big{(}(f^{*}f+p)^{-1}\big{)}{}^{-1}=\Phi_{\mathbbm{R}}\big{(}f(f^{*}f+p)^{-1}\big{)}\Phi_{\mathbbm{R}}\big{(}(f^{*}f+p)^{-1}\big{)}{}^{-1}=\Phi_{\mathbbm{R}}(f)
holds.
\boxempty
The continuous calculus from Proposition 5.8 now yields the universal
continuous calculus by taking a suitable quotient:
Lemma 5.10**.**
*Let A be a Su∗/̄algebra, N∈\mathbbmN and let \big{(}\mathbbm{R}^{N},\mathcal{I}_{\mathbbm{R}},\Phi_{\mathbbm{R}}\big{)} be the continuous
calculus for some pairwise commuting elements a1,…,aN∈AH from
Proposition 5.8. If f∈C(\mathbbmRN)H+ is coercive
and Φ\mathbbmR(f−1) invertible in A, then f∈I\mathbbmR.
*
Proof**:**
Given a coercive f∈C(\mathbbmRN)H+, then there exists ϵ∈]0,1] such that f≥ϵ\mathbbm1.
Write again p:=\mathbbm1+pr12+⋯+prN2∈Crt(\mathbbmRN)H+⊆(I\mathbbmR)H+
like in Proposition 5.8, then the pointwise estimates p≤ϵ−2f2p and f2≤ϵ−2f2p hold,
hence f2+p≤2ϵ−2f2p. Moreover, let h:=(2ϵ−2f2p)−1,
then 0≤h≤p−1 holds and therefore h∈C\SS(\mathbbmRN)H+ by the criterium of Lemma 5.5.
If Φ\mathbbmR(f−1) is invertible in A, then Φ\mathbbmR(h)=21ϵ2Φ\mathbbmR(f−1)2Φ\mathbbmR(p)−1
also is invertible in A, so f∈I\mathbbmR by part i.)
of Proposition 5.8 and because (f2+p)−1≥h.
\boxempty
Theorem 5.11**.**
Let A be a Su∗/̄algebra, N∈\mathbbmN and a1,…,aN∈AH pairwise commuting. Then the universal
continuous calculus \big{(}\operatorname{\mathrm{spec}}(a_{1},\dots,a_{N}),\mathcal{F}(a_{1},\dots,a_{N}),\Gamma_{a_{1},\dots,a_{N}}\big{)}
for a1,…,aN exists and can be constructed as follows:
i.)
The spectrum of a1,…,aN is given by
[TABLE]
2. ii.)
The intermediate ∗/̄subalgebra F(a1,…,aN) of \mathscr{C}\big{(}\operatorname{\mathrm{spec}}(a_{1},\dots,a_{N})\big{)}
is
[TABLE]
with I\mathbbmR the intermediate ∗/̄subalgebra of C(\mathbbmRN) from
Proposition 5.8, part i.).
3. iii.)
The unital ∗/̄homomorphism Γa1,…,aN:F(a1,…,aN)→A
is determined by
[TABLE]
for all f∈I\mathbbmR, with Φ\mathbbmR the unital ∗/̄homomorphism
Φ\mathbbmR:I\mathbbmR→A from
Proposition 5.8, part ii.).
*This construction has some additional properties: The map Γa1,…,aN:F(a1,…,aN)→A is
an injective positive unital ∗/̄homomorphism and even is an order embedding, and its image is a subset of the bicommutant
{a1,…,aN}′′. The space of functions F(a1,…,aN) is a commutative Su∗/̄algebra of finite type
with generators pr1,…,prN, and whenever f\in\mathscr{C}\big{(}\operatorname{\mathrm{spec}}(a_{1},\dots,a_{N})\big{)}{}^{+}_{\textup{H}} is coercive
and Γa1,…,aN(f−1) invertible in A, then f∈F(a1,…,aN).
*
Proof**:**
Let \big{(}\mathbbm{R}^{N},\mathcal{I}_{\mathbbm{R}},\Phi_{\mathbbm{R}}\big{)} be the continuous calculus for a1,…,aN that was constructed in
Proposition 5.8 and write again
p:=\mathbbm1+pr12+⋯+prN2∈Crt(\mathbbmRN)H+⊆(I\mathbbmR)H+.
As p is proper, I\mathbbmR is a proper Su∗/̄algebra of continuous functions on \mathbbmRN, so the results
from Section 4 apply: By Corollary 4.6,
the kernel of Φ\mathbbmR is a closed ∗/̄ideal of I\mathbbmR and can be described as
\ker\Phi_{\mathbbm{R}}=\mathcal{V}_{Z}=\big{\{}\,f\in\mathcal{I}_{\mathbbm{R}}\;\big{|}\;f|_{Z}=0\,\big{\}}, where Z is the closed subset
Z=\big{\{}\,x\in\mathbbm{R}^{N}\;\big{|}\;f(x)=0\textup{ for all }f\in\ker\Phi_{\mathbbm{R}}\,\big{\}} of \mathbbmRN
by Proposition 4.4.
In order to show that the universal continuous calculus for a1,…,aN exists and that it can be constructed
as described above, the first step is to show that Z coincides with the right-hand side of (5.4), i.e. that
[TABLE]
holds.
From \mathbbm1∈I\mathbbmR and prn∈I\mathbbmR for all n∈{1,…,N} it
follows that the continuous function fx,ϵ:=∑n=1N(xn\mathbbm1−prn)2−ϵ\mathbbm1
on \mathbbmRN is, for all x∈\mathbbmRN and all ϵ∈]0,∞[, an element of (I\mathbbmR)H.
Now given x∈\mathbbmRN and ϵ∈]0,∞[ such that ∑n=1N(xn\mathbbm1A−an)2≥ϵ\mathbbm1A,
then this means that Φ\mathbbmR(fx,ϵ)∈AH+
and Corollary 4.6 shows that fx,ϵ;−∈kerΦ\mathbbmR,
where fx,ϵ;− is the negative part of fx,ϵ.
In this case one has x∈/Z because fx,ϵ;−(x)=ϵ=0.
Conversely, if x∈\mathbbmRN\Z, then there exists ϵ∈]0,∞[ such that fx,ϵ(z)≥0
for all z∈Z because \mathbbmRN\Z is an open neighbourhood of x. So fx,ϵ and its positive part fx,ϵ;+
coincide on Z, and therefore
∑n=1N(xn\mathbbm1A−an)2−ϵ\mathbbm1A=Φ(fx,ϵ)=Φ(fx,ϵ;+)∈AH+,
which shows that ∑n=1N(xn\mathbbm1A−an)2 is coercive. We conclude that identity (∗ ‣ Proof) indeed holds.
The next step is to show that \mathcal{I}_{Z}\coloneqq\big{\{}\,f|_{Z}\;\big{|}\;f\in\mathcal{I}_{\mathbbm{R}}\,\big{\}} like in (5.5)
is an intermediate ∗/̄subalgebra of C(Z)
and that prZ;n∈IZ for all n∈{1,…,N}:
As the restriction map \ignorespaces⋅\ignorespaces∣Z:C(\mathbbmRN)→C(Z) is a unital ∗/̄homomorphism, it is clear
that IZ is a unital ∗/̄subalgebra of C(Z), and prZ;n=pr\mathbbmRN;n∣Z∈IZ
holds for all n∈{1,…,N}. In order to check that C(Z)bd⊆IZ,
let a Hermitian element f∈C(Z)Hbd be given. Then one can construct an extension
fext∈C(\mathbbmRN)Hbd⊆I\mathbbmR fulfilling
fext∣Z=f by applying Tietze’s extension theorem, so f∈IZ.
For general (not necessarily Hermitian) f∈C(Z)bd this shows that Re(f),Im(f)∈IZ
and therefore also f∈IZ.
Note that instead of the most general form of Tietze’s extension theorem, one can also use one
of the well-known explicit formulae for such an extension of a continuous function f:Z→\mathbbmR which is bounded from below.
An example is
[TABLE]
for all x∈\mathbbmRN\Z, where d is any metric on \mathbbmRN that induces its standard topology
and dZ:\mathbbmRN→[0,∞[ the distance from Z, i.e. dZ(x):=infz∈Zd(z,x) for all x∈\mathbbmRN.
Like in (5.6) one can now construct a well-defined unital ∗/̄homomorphism ΦZ:IZ→A,
ΦZ(f∣Z):=Φ\mathbbmR(f),
because \ker\Phi_{\mathbbm{R}}\supseteq\big{\{}\,f\in\mathcal{I}_{\mathbbm{R}}\;\big{|}\;f|_{Z}=0\,\big{\}} as shown above.
Then ΦZ(prZ;n)=Φ\mathbbmR(pr\mathbbmRN;n)=an holds for all n∈{1,…,N},
so (Z,IZ,ΦZ) is a continuous calculus for a1,…,aN. It is even
the universal one:
Let any continuous calculus (Y,J,Ψ) for a1,…,aN be given. Then Y⊇Z
because for every x∈\mathbbmRN\Y there is an ϵ∈]0,∞[ such that fx,ϵ∣Y≥0
as discussed above, and therefore ∑n=1N(xn\mathbbm1A−an)2−ϵ\mathbbm1A=Ψ(fx,ϵ∣Y)≥0,
which shows that x∈\mathbbmRN\Z by (∗ ‣ Proof). It is now easy to check that \big{(}\mathbbm{R}^{N},\mathcal{J}^{\mathrm{ext}},\Psi^{\mathrm{ext}}\big{)}
with \mathcal{J}^{\mathrm{ext}}\coloneqq\big{\{}\,f^{\mathrm{ext}}\in\mathscr{C}(\mathbbm{R}^{N})\;\big{|}\;f^{\mathrm{ext}}|_{Y}\in\mathcal{J}\,\big{\}} and
Ψext(f):=Ψ(fext∣Y) for all fext∈Jext is a continuous
calculus for a1,…,aN, so Corollary 5.9 shows that Jext⊆I\mathbbmR
and Ψext(fext)=Φ\mathbbmR(fext) for all fext∈Jext.
Given any f∈JH+, then one can again construct an extension
fext∈C(\mathbbmRN)H+ fulfilling fext∣Y=f
by using the explicit formula (∗∗ ‣ Proof) with Y in place of Z, or by taking the positive part of any extension of f to \mathbbmRN.
But then
fext∈Jext⊆I\mathbbmR implies that f∣Z=fext∣Z∈IZ
and ΦZ(f∣Z)=Φ\mathbbmR(fext)=Ψext(fext)=Ψ(f). As a general
f∈J can be decomposed as f=∑k=03ikfk with fk∈JH+,
e.g. using the positive and negative parts of the real and imaginary parts of f, this guarantees
that f∣Z∈IZ and ΦZ(f∣Z)=Ψ(f) hold for all f∈J. So (Z,IZ,ΦZ) is the
universal continuous calculus for a1,…,aN, thus spec(a1,…,aN)=Z, F(a1,…,aN)=IZ
and Γa1,…,aN=ΦZ, and (5.4), (5.5) and (5.6)
are fulfilled.
It only remains to check the additional properties mentioned in the statement of this theorem:
By construction of ΦZ, its image in A is identical to the image of Φ\mathbbmR, hence is again a subset
of the bicommutant {a1,…,aN}′′, see Proposition 5.8.
Being a unital ∗/̄homomorphism defined on a proper Su∗/̄algebra of continuous functions, ΦZ
is automatically positive by Corollary 4.6. It is also injective because
ΦZ(f∣Z)=0 for some f∈I\mathbbmR means Φ\mathbbmR(f)=0, i.e. f∈kerΦ\mathbbmR,
which means f∣Z=0 as discussed above. Corollary 4.6 now also shows that ΦZ is
an order embedding, because if ΦZ(f)∈AH+ with some f∈(IZ)H,
then ΦZ(f)=ΦZ(f+) implies f=f+≥0.
As I\mathbbmR is a commutative
Su∗/̄algebra of finite type with generators pr\mathbbmRN;1,…,pr\mathbbmRN;N by
Proposition 5.8, and as IZ is
the image of I\mathbbmR under the positive unital ∗/̄homomorphism \ignorespaces⋅\ignorespaces∣Z,
it is also a commutative Su∗/̄algebra of finite type with generators prZ;1,…,prZ;N
by Proposition 3.12.
Finally, if f∈C(Z)H+ is coercive, then there exists ϵ∈]0,∞[ such that f(z)≥ϵ
for all z∈Z and one can construct an extension fext∈C(\mathbbmRN)H+ fulfilling fext∣Z=f and
fext(x)≥ϵ for all x∈\mathbbmRN by using the explicit formula (∗∗ ‣ Proof)
or by taking the pointwise maximum with ϵ of any extension of f to \mathbbmRN. Its inverse
(fext)−1∈C(\mathbbmRN)bd⊆I\mathbbmR then fulfils (fext)−1∣Z=f−1, thus
\Phi_{\mathbbm{R}}\big{(}(f^{\mathrm{ext}})^{-1}\big{)}=\Phi_{Z}(f^{-1}).
If ΦZ(f−1) is invertible in A, then it follows from the previous Lemma 5.10 that fext∈I\mathbbmR
and therefore f∈IZ.
\boxempty
Note that this allows to construct e.g. absolute values and positive or negative parts
of Hermitian elements of a Su∗/̄algebra A, and also square roots of positive Hermitian elements and
“suprema” or “infima” of commuting Hermitian elements by application of the universal continuous
calculus to the corresponding function. The result is the same as the constructions
in [14] because the algebraic and order theoretic properties match,
as one can easily check: For example, given a∈AH+, then (x\mathbbm1−a)2≥x2\mathbbm1
for all x∈]−∞,0[ shows that spec(a)⊆[0,∞[, and \Gamma_{a}\big{(}\sqrt{\ignorespaces{\,\cdot\,}\ignorespaces}|_{\operatorname{\mathrm{spec}}(a)}\big{)}
yields the square root of a because it is a positive Hermitian element of the bicommutant {a}′′ that squares to a.
The universal continuous calculus applies especially to the Su∗/̄algebras of
operators on a Hilbert space as in [14, Sec. 8]
that can be constructed out of any selfadjoint operator. The spectrum of a single Hermitian element
can then be described in the usual way:
Corollary 5.12**.**
Let A be a Su∗/̄algebra and a∈AH, then
[TABLE]
Proof**:**
If λ∈\mathbbmR has the property that λ\mathbbm1−a is invertible in A
with uniformly bounded inverse, then (λ\mathbbm1−a)−2≤∥(λ\mathbbm1−a)−1∥∞2\mathbbm1
implies ∥(λ\mathbbm1−a)−1∥∞−2\mathbbm1≤(λ\mathbbm1−a)2, so λ∈\mathbbmR\spec(a)
by the previous Theorem 5.11.
Conversely, let pr:=id\mathbbmR∣spec(a)∈F(a) be the coordinate function
and let λ∈\mathbbmR\spec(a) be given, then λ\mathbbm1F(a)−pr
is invertible in C(spec(a)) and its inverse is uniformly bounded because spec(a) is closed,
so (λ\mathbbm1F(a)−pr)−1∈C(spec(a))bd⊆F(a).
It follows that \lambda\mathbbm{1}_{\mathcal{A}}-a=\Gamma_{a}\big{(}\lambda\mathbbm{1}_{\mathcal{F}(a)}-\mathrm{pr}\big{)} is invertible in A
with uniformly bounded inverse.
\boxempty
A normal element of a ∗/̄algebra is an element a fulfilling aa∗=a∗a. This is the case
if and only if its real and imaginary parts Re(a) and Im(a) commute. By identifying \mathbbmR2 with \mathbbmC via
the \mathbbmR-linear isomorphism \mathbbmR2∋(x1,x2)↦x1+ix2∈\mathbbmC and interpreting
\operatorname{\mathrm{spec}}_{\mathbbm{C}}(a)\coloneqq{\operatorname{\mathrm{spec}}}\big{(}\mathsf{Re}(a),\mathsf{Im}(a)\big{)} as a closed subset of \mathbbmC, one obtains
the usual description of the spectrum of normal elements and the spectral mapping theorem:
Corollary 5.13**.**
Let A be a Su∗/̄algebra and a∈A a normal element, then
[TABLE]
*Moreover, given pairwise commuting elements a1,…,aN∈AH with N∈\mathbbmN and
f∈F(a1,…,aN), then {\operatorname{\mathrm{spec}}_{\mathbbm{C}}}\big{(}\Gamma_{a_{1},\dots,a_{N}}(f)\big{)}
is the closure in \mathbbmC of the image of spec(a1,…,aN) under f.
*
Proof**:**
If λ∈\mathbbmC has the property that λ\mathbbm1−a is invertible in A
with uniformly bounded inverse, then ((λ\mathbbm1−a)−1)∗(λ\mathbbm1−a)−1≤∥(λ\mathbbm1−a)−1∥∞2\mathbbm1
implies
Conversely, \mathrm{id}_{\mathbbm{C}}|_{\operatorname{\mathrm{spec}}_{\mathbbm{C}}(a)}=\mathrm{pr}_{1}+\mathrm{i}\,\mathrm{pr}_{2}\in\mathcal{F}\big{(}\mathsf{Re}(a),\mathsf{Im}(a)\big{)},
and for λ∈\mathbbmC\spec\mathbbmC(a) one thus finds that
λ\mathbbm1−(pr1+ipr2) is invertible in \mathscr{C}\big{(}\operatorname{\mathrm{spec}}_{\mathbbm{C}}(a)\big{)}
with uniformly bounded inverse because spec\mathbbmC(a) is closed, so
\big{(}\lambda\mathbbm{1}-(\mathrm{pr}_{1}+\mathrm{i}\,\mathrm{pr}_{2})\big{)}{}^{-1}\in\mathscr{C}\big{(}\operatorname{\mathrm{spec}}_{\mathbbm{C}}(a)\big{)}{}^{\mathrm{bd}}\subseteq\mathcal{F}\big{(}\mathsf{Re}(a),\mathsf{Im}(a)\big{)}
and \lambda\mathbbm{1}-a=\Gamma_{\mathsf{Re}(a),\mathsf{Im}(a)}\big{(}\lambda\mathbbm{1}-(\mathrm{pr}_{1}+\mathrm{i}\,\mathrm{pr}_{2})\big{)} is invertible in A
with inverse \Gamma_{\mathsf{Re}(a),\mathsf{Im}(a)}\big{(}(\lambda\mathbbm{1}-(\mathrm{pr}_{1}+\mathrm{i}\,\mathrm{pr}_{2}))^{-1}\big{)}\in\mathcal{A}^{\mathrm{bd}}.
Finally, let pairwise commuting a1,…,aN∈AH with N∈\mathbbmN and f∈F(a1,…,aN)
be given. For λ∈\mathbbmC define fλ:=λ\mathbbm1−f∈F(a1,…,aN),
then for every ϵ∈]0,∞[, one has fλ∗fλ≥ϵ2\mathbbm1F(a1,…,aN) if and only
if Γa1,…,aN(fλ∗fλ)≥ϵ2\mathbbm1A because Γa1,…,aN
is an injective positive unital ∗/̄homomorphism and an order embedding by Theorem 5.11.
As A is a Su∗/̄algebra and especially is symmetric, Γa1,…,aN(fλ∗fλ)≥ϵ2\mathbbm1A
is equivalent to Γa1,…,aN(fλ) being invertible in A
with inverse fulfilling \big{(}\Gamma_{a_{1},\dots,a_{N}}(f_{\lambda})^{-1}\big{)}^{*}\Gamma_{a_{1},\dots,a_{N}}(f_{\lambda})^{-1}\leq\epsilon^{-2}\mathbbm{1}_{\mathcal{A}},
i.e. ∥Γa1,…,aN(fλ)−1∥∞≤ϵ−1.
It is now easy to check that, for fixed λ∈\mathbbmC, there exists ϵ∈]0,∞[
such that fλ∗fλ≥ϵ2\mathbbm1F(a1,…,aN) if and only
λ is not an element of the closure of the image of spec(a1,…,aN) under f.
By the above considerations, this is equivalent to
λ\mathbbm1A−Γa1,…,aN(f) being invertible in A
with inverse fulfilling ∥(λ\mathbbm1A−Γa1,…,aN(f))−1∥∞=∥Γa1,…,aN(fλ)−1∥∞≤ϵ−1
for this ϵ∈]0,∞[,
i.e. to \lambda\in\mathbbm{C}\backslash{\operatorname{\mathrm{spec}}_{\mathbbm{C}}}\big{(}\Gamma_{a_{1},\dots,a_{N}}(f)\big{)} by (5.8).
\boxempty
Note that the universal continuous calculus
for a normal element a of a Su∗/̄algebra yields the expected results
ΓRe(a),Im(a)(id\mathbbmC∣spec\mathbbmC(a))=ΓRe(a),Im(a)(pr1+ipr2)=a
and ΓRe(a),Im(a)(\ignorespaces⋅\ignorespaces∣spec\mathbbmC(a))=ΓRe(a),Im(a)(pr1−ipr2)=a∗.
6 Representations of commutative Su∗/̄algebras of finite type
Finally, the universal continuous calculus allows to identify the commutative Su∗/̄algebras of finite type
as the proper Su∗/̄algebras of continuous functions on closed subsets of Euclidean space:
Proposition 6.1**.**
*Every proper Su∗/̄algebra of continuous functions on a closed subset X of some \mathbbmRN with N∈\mathbbmN
is a commutative Su∗/̄algebra of finite type.
*
Proof**:**
Let X be a closed subset of \mathbbmRN for some N∈\mathbbmN and let I⊆C(X) be a proper
Su∗/̄algebra of continuous functions on X. We have to find generators for I:
First assume that X is compact, then I=C(X)=C(X)bd and especially prn∈I holds
for all n∈{1,…,N}. In this case I is a commutative Su∗/̄algebra of finite type with generators
pr1,…,prN, because p:=\mathbbm1+pr12+⋯+prN2 is clearly proper in I
and because ⟨⟨{p−1,pr1p−1,…,prNp−1}⟩⟩C∗=I by the Stone-Weierstraß Theorem.
Otherwise, i.e. if X is non-compact, it is possible that prn∈/I for some n∈{1,…,N} and the construction of generators
is more complicated: In this case let p∈IH+ be a proper function and define
\lVert x\rVert\coloneqq\max\big{\{}\lvert x_{1}\rvert,\dots,\lvert x_{N}\rvert\big{\}} for all x∈X. Then for every k∈\mathbbmN0
there exists x∈X with ∥x∥≥k and the sequence
\mathbbmN0∋k↦p~k:=infx∈X,∥x∥≥kp(x)∈[0,∞[
is increasing and unbounded. One can now construct an unbounded sequence (αk)k∈\mathbbmN0
that starts with α0=0 and which fulfils αk−1<αk and αk≤1+p~k−1
for all k∈\mathbbmN; for example, αk:=1−2−k+p~k−1 for all k∈\mathbbmN is a suitable choice.
From this sequence (αk)k∈\mathbbmN one obtains a piecewise defined continuous function g:\mathbbmR→\mathbbmR,
[TABLE]
fulfilling g(−t)=−g(t) for all t∈\mathbbmR and
∣g(xn)∣=g(∣xn∣)≤αk≤1+p~k−1≤1+p(x) for all x∈X
and all n∈{1,…,N} and for a suitable k∈\mathbbmN such that k−1≤∣xn∣≤k.
This function g
is strictly increasing, i.e. g(t)<g(t′) for all t,t′∈\mathbbmR with t<t′, hence injective
and open. Moreover, limt→±∞g(t)=±∞ holds, so g is also surjective and therefore
is a homeomorphism. As a consequence, the map
g×N:\mathbbmRN→\mathbbmRN, (x_{1},\dots,x_{N})\mapsto\big{(}g(x_{1}),\dots,g(x_{N})\big{)} is a homeomorphism, too.
Now define Y as the image of X under g×N, which is a closed subset
of \mathbbmRN because g×N is a homeomorphism. Define
ϕ:=X→Y as ϕ(x):=g×N(x) for all x∈X, which is a well-defined homeomorphism,
and \mathcal{J}\coloneqq\big{\{}\,f\in\mathscr{C}(Y)\;\big{|}\;f\circ\phi\in\mathcal{I}\,\big{\}}. Then it is easy to check
that J is an intermediate ∗/̄subalgebra of C(Y) and that the map
Φ:J→I, f↦Φ(f):=f∘ϕ
is an isomorphism of ordered ∗/̄algebras whose inverse is given by I∋f↦f∘ϕ−1∈J.
Moreover, prY;n∈J for all n∈{1,…,N}
because ∣prY;n∘ϕ∣=∣g∘prX;n∣≤\mathbbm1I+p shows that prY;n∘ϕ∈I by Proposition 3.2.
So (Y,J,Φ) is a continuous calculus for g∘prX;1,…,g∘prX;N∈IH
and Φ maps surjectively onto I. It follows that the universal continuous
calculus for g∘prX;1,…,g∘prX;N also maps surjectively onto I,
hence is an isomorphism (one actually finds that (Y,J,Φ) is the universal continuous calculus
because Φ is an isomorphism).
By Theorem 5.11, I≅F(g∘prX;1,…,g∘prX;N)
is a commutative Su∗/̄algebra of finite type and g∘prX;1,…,g∘prX;N are generators of I.
\boxempty
The converse is also true:
Theorem 6.2**.**
*For every commutative Su∗/̄algebra of finite type A with
generators a1,…,aN∈AH, N∈\mathbbmN, the universal continuous calculus
Γa1,…,aN:F(a1,…,aN)→A is an isomorphism of ordered ∗/̄algebras,
and thus allows to identify A with the proper Su∗/̄algebra of continuous functions F(a1,…,aN)
on the closed subset spec(a1,…,aN) of \mathbbmRN.
*
Proof**:**
Let A be a commutative Su∗/̄algebra of finite type with generators a1,…,aN∈AH, N∈\mathbbmN,
so p:=\mathbbm1+a12+⋯+aN2 is proper in A. By Theorem 5.11, the universal continuous
calculus for a1,…,aN exists and Γa1,…,aN:F(a1,…,aN)→A
is an injective unital ∗/̄homomorphism and even an order embedding. We have to show that Γa1,…,aN
is also surjective:
Given b∈A, then there are numerator and denominator
bn′,bd′∈⟨⟨{p−1,a1p−1,…,aNp−1}⟩⟩C∗
such that bd′ is invertible in A and b=bn′(bd′)−1.
By setting bn:=bn′(bd′)∗,bd:=bd′(bd′)∗
one obtains numerator and denominator
bn,bd∈⟨⟨{p−1,a1p−1,…,aNp−1}⟩⟩C∗
such that bd is positive Hermitian and invertible in A and b=bnbd−1.
By Proposition 5.7 and the subsequent construction of the
universal continuous calculus in Proposition 5.8 and Theorem 5.11,
the C∗/̄subalgebra ⟨⟨{p−1,a1p−1,…,aNp−1}⟩⟩C∗
of Abd lies in the image of Γa1,…,aN, so there especially exist
fn,fd∈F(a1,…,aN) such that Γa1,…,aN(fn)=bn
and Γa1,…,aN(fd)=bd. As Γa1,…,aN is
injective and an order embedding, fd is positive Hermitian and uniformly bounded.
Moreover, even fd(x)>0 holds for all x∈spec(a1,…,aN):
Assume to the contrary that there is some x^∈spec(a1,…,aN) for which fd(x^)=0.
In this case, let p~:=\mathbbm1+pr12+⋯+prN2∈F(a1,…,aN)H+
and λ:=p~(x^)+1∈[1,∞[, then
Γa1,…,aN(p~)=p is proper in A and therefore
there are μ∈[0,∞[ and c∈AH+ such that for every ϵ∈]0,∞[
there is a k∈\mathbbmN0 for which bd−1≤μ\mathbbm1A+(p/λ)k+ϵc holds.
Construct g∈F(a1,…,aN)H+ as the positive part of
\frac{1}{2}\big{(}\mathbbm{1}-2(\mu+1)f_{\mathrm{d}}-\tilde{p}/\lambda\big{)},
then 0≤g≤\mathbbm1/2, g(x^)=(2λ)−1>0 and g(x)=0 holds for all x∈spec(a1,…,aN) for which
p~(x)≥λ and also for all x∈spec(a1,…,aN) for which (μ+1)fd(x)≥1/2.
Multiplying the inequality bd−1≤μ\mathbbm1+(p/λ)k+ϵc with bd=(bd′)∗bd′
yields \mathbbm{1}\leq\big{(}\mu\mathbbm{1}+(p/\lambda)^{k}\big{)}b_{\mathrm{d}}+\epsilon b_{\mathrm{d}}c, and as \mathbbm1, bd
and p have (necessarily unique) preimages under Γa1,…,aN, one obtains the estimate
[TABLE]
and therefore also
[TABLE]
where the first inequality follows from the estimate
g^{2}\leq\big{(}\mathbbm{1}-\big{(}\mu\mathbbm{1}+(\tilde{p}/\lambda)^{k}\big{)}f_{\mathrm{d}}\big{)}g
that can easily be checked pointwise for x∈spec(a1,…,aN)
by using that g(x)=0 only if \big{(}\mathbbm{1}-f_{\mathrm{d}}\big{(}\mu\mathbbm{1}+(\tilde{p}/\lambda)^{k}\big{)}\big{)}(x)>1/2.
As g is independent of k, this estimate Γa1,…,aN(g2)≤ϵbdcΓa1,…,aN(g)
holds for all ϵ∈]0,∞[, so Γa1,…,aN(g2)=0 because A is Archimedean,
therefore g2=0 because Γa1,…,aN is injective. But this contradicts g(x^)>0, so the assumption
that there exists x^∈spec(a1,…,aN) with fd(x^)=0 has to be false.
We thus have seen that fd(x)>0 holds for all x∈spec(a1,…,aN), which means that the pointwise inverse
f_{\mathrm{d}}^{-1}\in\mathscr{C}\big{(}\operatorname{\mathrm{spec}}(a_{1},\dots,a_{N})\big{)}^{+}_{\textup{H}} exists. Even fd−1∈F(a1,…,aN)
holds by the criterium from Theorem 5.11 because the inverse fd of fd−1 is uniformly
bounded and Γa1,…,aN(fd)=bd invertible in A. It follows that Γa1,…,aN(fnfd−1)=bnbd−1=b
and we conclude that Γa1,…,aN:F(a1,…,aN)→A is surjective, hence an isomorphism of ordered ∗/̄algebras.
\boxempty
In the above proof, the assumption of properness enters in an essential, but perhaps not very intuitiv way. A short example therefore might be helpful:
Example 6.3**.**
In the commutative Su∗/̄algebra \mathscr{C}\big{(}{]0,1]}\big{)} the function \mathrm{id}_{]0,1]}\in\mathscr{C}\big{(}{]0,1]}\big{)}{}^{\mathrm{bd}} alone is not a sufficient tuple of generators
because spec(id]0,1])=[0,1] by Corollary 5.12, so that \mathcal{F}(\mathrm{id}_{]0,1]})=\mathscr{C}\big{(}[0,1]\big{)}=\mathscr{C}\big{(}[0,1]\big{)}{}^{\mathrm{bd}}
is uniformly bounded and its image under Γid]0,1] is only a unital ∗/̄subalgebra of \mathscr{C}\big{(}{]0,1]}\big{)}{}^{\mathrm{bd}}.
While one can express every f\in\mathscr{C}\big{(}{]0,1]}\big{)} as a quotient f=fnfd−1
with fn,fd in the image of Γid]0,1] and fd invertible in \mathscr{C}\big{(}{]0,1]}\big{)},
e.g. fd=(f∗f+(id]0,1])−1)−1 and fn=f(f∗f+(id]0,1])−1)−1,
the preimage of such an invertible fd under Γid]0,1] need not be invertible:
Indeed, id[0,1]∈F(id]0,1]) is the preimage of \mathrm{id}_{]0,1]}\in\mathscr{C}\big{(}{]0,1]}\big{)}.
This all relates to \mathbbm{1}+(\mathrm{id}_{]0,1]})^{2}\in\mathscr{C}\big{(}{]0,1]}\big{)} not being proper.
As a final remark we note that Proposition 4.7 is especially interesting in connection with the above Theorem 6.2:
Let A and B be two commutative Su∗/̄algebras of finite type with generators a1,…,aM∈AH
and b1,…,bN∈BH, M,N∈\mathbbmN, which thus are isomorphic as ordered ∗/̄algebras
to the proper Su∗/̄algebras of continuous functions F(a1,…,aM) and F(b1,…,bN)
via the isomorphisms Γa1,…,aM and Γb1,…,bN, respectively.
Then for every unital ∗/̄homomorphism Φ:A→B there exists a unique continuous map
ϕ:spec(b1,…,bN)→spec(a1,…,aM) such that f∘ϕ∈F(b1,…,bN) and
\Phi\big{(}\Gamma_{a_{1},\dots,a_{M}}(f)\big{)}=\Gamma_{b_{1},\dots,b_{N}}(f\circ\phi) hold for all f∈F(a1,…,aM).
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