C*-Algebras for partial product systems over N

TL;DR

This paper introduces partial product systems over N, generalizing existing structures, and develops associated Toeplitz and Cuntz-Pimsner C*-algebras, linking them to Fell bundles over Z and analyzing their ideal structure.

Contribution

It defines partial product systems over N and constructs their Toeplitz and Cuntz-Pimsner algebras, extending the theory of product systems and Fell bundles.

Findings

Section C*-algebra of a Fell bundle over Z is a relative Cuntz-Pimsner algebra.

Characterization of gauge-invariant ideals in Toeplitz C*-algebra.

Generalization of product systems and Fell bundles over N and Z.

Abstract

We define partial product systems over N. They generalise product systems over N and Fell bundles over Z. We define Toeplitz C*-algebras and relative Cuntz-Pimsner algebras for them and show that the section C*-algebra of a Fell bundle over Z is a relative Cuntz-Pimsner algebra. We describe the gauge-invariant ideals in the Toeplitz C*-algebra.

Equations

$$S_{n,m}(x):{\mathcal{E}}_m\to {\mathcal{E}}_{n+m},y\mapsto {\mu }_{n,m}(x\otimes y).$$

$${\omega }_n(x{)}^{\ast }{\omega }_m(y)=\{\begin{array}{ll}{\omega }_{m-n}(S_n(x{)}^{\ast }y)& \text{if }m>n,\\ {\omega }_{n-m}(S_m(y{)}^{\ast }x{)}^{\ast }& \text{if }n>m,\\ {\omega }_0(⟨y|x⟩{)}^{\ast }& \text{if }n=m;\end{array}$$

$$S_n(x{)}^{\ast }{S}_m(y)=S_{m-n}(S_n(x{)}^{\ast }y)$$

$${\Theta }_{m,n}:\mathbb{K}({\mathcal{E}}_m,{\mathcal{E}}_n)\to \mathcal{T}$$

$${\Theta }_{m,n}(|x⟩⟨y|)={\bar{\omega }}_n(x){\bar{\omega }}_m(y{)}^{\ast }$$

$$⟨x_1\otimes y_1|x_2\otimes y_2{⟩}_C:=⟨y_1|⟨x_1|x_2{⟩}_A\cdot y_2{⟩}_B$$

$$(\mathcal{E}{\otimes }_B\mathcal{F}){\otimes }_C\mathcal{G}$$

$$\mathcal{A}{\otimes }_A\mathcal{E}$$

$$\mathcal{E}{\otimes }_B\mathcal{B}$$

$${\mathcal{E}}_n{\otimes }_A{\mathcal{E}}_m{\otimes }_A{\mathcal{E}}_l$$

$${\Theta }_{m,n}(|x⟩⟨y|)={\omega }_n(x){\omega }_m(y{)}^{\ast }$$

$${\Theta }_{m,n}(S)\cdot {\Theta }_{l,m}(T)={\Theta }_{l,n}(S\cdot T),{\Theta }_{n,m}(S^{\ast })={\Theta }_{m,n}(S{)}^{\ast }$$

$${\omega }_n(x){\omega }_m(y{)}^{\ast }{\omega }_m(z){\omega }_l(w{)}^{\ast }={\omega }_n(x){\omega }_0(⟨y|z⟩){\omega }_l(w{)}^{\ast }={\omega }_n(x\cdot ⟨y|z⟩){\omega }_l(w{)}^{\ast }.$$

$${\Theta }_{n,0}(x)={\omega }_n(x),{\Theta }_{0,n}(x^{\ast })={\omega }_n(x{)}^{\ast }$$

$${\Theta }_{m,n}(T){\omega }_m(x)={\omega }_n(T(x))$$

$$\ker {\omega }_n={\mathcal{E}}_n\cdot I\subseteq {\mathcal{E}}_n,\ker {\Theta }_{m,n}=\mathbb{K}({\mathcal{E}}_m,{\mathcal{E}}_n\cdot I)\subseteq \mathbb{K}({\mathcal{E}}_m,{\mathcal{E}}_n).$$

$$S_{n,m}(x):{\mathcal{E}}_m\to {\mathcal{E}}_{n+m},y\mapsto x\cdot y:={\mu }_{n,m}(x\otimes y).$$

$$⟨x|y⟩(v):=\sum _{e\in s^{-1}(v)}\overline{x(e)}y(e)$$

$$⟨{\delta }_x|{\delta }_y{⟩}_A=\{\begin{array}{ll}{\delta }_{s_n(x)}& \text{if }x=y,\\ 0& \text{otherwise;}\end{array}$$

$$(x\cdot a)(e):=x(e)\cdot a(s(e)),(a\cdot x)(e):=a(r(e))\cdot x(e)$$

$${\mu }_{n,m}({\delta }_x\otimes {\delta }_y):=\{\begin{array}{ll}{\delta }_{x\cdot y}& \text{if }{s}_n(x)=r_m(y),\\ 0& \text{otherwise.}\end{array}$$

$$\big{\langle}\mu_{n,m}(\delta_{x}\otimes\delta_{y})\,\big{|}\,\mathopen{}\mu_{n,m}(\delta_{x^{\prime}}\otimes\delta_{y^{\prime}})\big{\rangle}(v)=\langle\delta_{x\cdot y}\,|\,\mathopen{}\delta_{x^{\prime}\cdot y^{\prime}}\rangle(v)\\ =\delta_{s(x)=r(y)}\cdot\delta_{s(x^{\prime})=r(y^{\prime})}\cdot\delta_{x\cdot x^{\prime}=y\cdot y^{\prime}}\cdot\delta_{s(x\cdot y)=v}.$$

$$\big{\langle}\delta_{x}\otimes\delta_{y}\,\big{|}\,\mathopen{}\delta_{x^{\prime}}\otimes\delta_{y^{\prime}}\big{\rangle}(v)=\big{\langle}\delta_{y}\,\big{|}\,\mathopen{}\langle\delta_{x}\,|\,\mathopen{}\delta_{x^{\prime}}\rangle\delta_{y^{\prime}}\rangle(v)=\delta_{x=x^{\prime}}\langle\delta_{y}\,|\,\mathopen{}\delta_{s(x)}\delta_{y^{\prime}}\rangle(v)\\ =\delta_{x=x^{\prime}}\delta_{s(x)=r(y^{\prime})}\langle\delta_{y}\,|\,\mathopen{}\delta_{y^{\prime}}\rangle(v)=\delta_{x=x^{\prime}}\delta_{s(x)=r(y^{\prime})}\delta_{y=y^{\prime}}\delta_{s(y)=v}.$$

$$(x\cdot y)\cdot z=x\cdot (y\cdot z)$$

$$S_{{\delta }_x}^{\ast }({\delta }_y):=\{\begin{array}{ll}z& \text{if }y=x\cdot z\text{ for some }z\in E,\\ 0& \text{otherwise.}\end{array}$$

$${\Theta }_{m,n}(\mathbb{K}({\mathcal{E}}_m,{\mathcal{E}}_n))\cdot {\Theta }_{p,q}(\mathbb{K}({\mathcal{E}}_p,{\mathcal{E}}_q))\subseteq \{\begin{array}{ll}{\Theta }_{m-q+p,n}(\mathbb{K}({\mathcal{E}}_{m-q+p},{\mathcal{E}}_n))& \text{if }m\ge q,\\ {\Theta }_{p,n+q-m}(\mathbb{K}({\mathcal{E}}_p,{\mathcal{E}}_{n+q-m}))& \text{if }m\le q.\end{array}$$

$$\text{ref}simplifiabilit{y}_1⟹\text{ref}simplifiabilit{y}_3⟹\text{ref}simplifiabilit{y}_5,\text{ref}simplifiabilit{y}_2⟹\text{ref}simplifiabilit{y}_4⟹\text{ref}simplifiabilit{y}_6$$

$$\begin{gathered} {\omega }_{m-n}(z{)}^{\ast }{\omega }_{m-n}(S_n(x{)}^{\ast }(y))={\omega }_0(⟨z|S_n(x{)}^{\ast }(y)⟩)={\omega }_0(⟨S_n(x)(z)|y⟩) \\ ={\omega }_0(⟨x\cdot z|y⟩)={\omega }_m(x\cdot z{)}^{\ast }{\omega }_m(y)={\omega }_{m-n}(z{)}^{\ast }{\omega }_n(x{)}^{\ast }{\omega }_m(y). \end{gathered}$$

$$X:={\omega }_{m-n}(S_n(x{)}^{\ast }(y))-{\omega }_n(x{)}^{\ast }{\omega }_m(y),$$

$${\omega }_n({\mathcal{E}}_n)={\Theta }_{0,n}(\mathbb{K}({\mathcal{E}}_0,{\mathcal{E}}_n)),{\omega }_n({\mathcal{E}}_n{)}^{\ast }={\Theta }_{n,0}(\mathbb{K}({\mathcal{E}}_n,{\mathcal{E}}_0)).$$

$${\Theta }_{m,n}(\mathbb{K}({\mathcal{E}}_m,{\mathcal{E}}_n))={\omega }_n({\mathcal{E}}_n){\omega }_m({\mathcal{E}}_m{)}^{\ast }.$$

Full text

This article is based on the Master’s Thesis of the second author.

1. Introduction

The Cuntz–Pimsner algebras introduced by Pimsner in [Pimsner:Generalizing_Cuntz-Krieger] were generalised, among others, by Muhly and Solel [Muhly-Solel:Tensor] and by Katsura [Katsura:Cstar_correspondences]. Fowler [Fowler:Product_systems] generalised Pimsner’s construction to product systems. A self-correspondence $\mathcal{E}$ of a $\mathrm{C}^{*}$-algebra $A$ generates a product system over $\mathbb{N}$ by taking $\mathcal{E}_{n}\mathrel{\vcentcolon=}\mathcal{E}^{\otimes_{A}n}$ with the obvious multiplication maps $\mu_{n,m}\colon\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}\to\mathcal{E}_{n+m}$ for $n,m\in\mathbb{N}$. Any product system over $\mathbb{N}$ is isomorphic to one that is built from a $\mathrm{C}^{*}$-correspondence like this. Another source of product systems over $\mathbb{N}$ are Fell bundles over $\mathbb{Z}$ (see [Exel:Partial_dynamical]). They consist of Hilbert bimodules $(\mathcal{E}_{n})_{n\in\mathbb{Z}}$ with involutions $\mathcal{E}_{n}\cong\mathcal{E}_{-n}^{*}$, $x\mapsto x^{*}$, and multiplication maps $\mu_{n,m}\colon\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}\to\mathcal{E}_{n+m}$, now for all $n,m\in\mathbb{Z}$. Due to the involutions, the Hilbert bimodules $\mathcal{E}_{n}$ for $n\in\mathbb{N}$ with the multiplication maps $\mu_{n,m}$ for $n,m\in\mathbb{N}$ suffice to recover the entire Fell bundle. This data gives a product system over $\mathbb{N}$ if and only if the maps $\mu_{n,m}$ are surjective and hence unitary for all $n,m\in\mathbb{N}$. Then the Fell bundle is called semi-saturated. In general, the multiplication maps are only isometries of Hilbert bimodules, possibly without adjoint.

The construction of a (relative) Cuntz–Pimsner algebra of a product system splits into two steps. The first builds a Fell bundle over $\mathbb{Z}$, the second takes the section $\mathrm{C}^{*}$-algebra of that Fell bundle. This viewpoint is used in [Meyer-Sehnem:Bicategorical_Pimsner] to interpret relative Cuntz–Pismner algebras in bicategorical terms. So there is a close and important link between product systems over $\mathbb{N}$ and Fell bundles over $\mathbb{Z}$. This article describes a common generalisation for both, which we call partial product systems.

A partial product system over $\mathbb{N}$ consists of a $\mathrm{C}^{*}$-algebra $A$ with $A,A$-correspondences $\mathcal{E}_{n}$ for all $n\in\mathbb{N}$ and isometries $\mu_{n,m}\colon\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}\hookrightarrow\mathcal{E}_{n+m}$ for all $n,m\in\mathbb{N}$, subject to several conditions. The obvious conditions are that the multiplication maps $\mu_{n,m}$ be associative, that $\mathcal{E}_{0}=A$ and that $\mu_{0,n}\colon A\otimes_{A}\mathcal{E}_{n}\hookrightarrow\mathcal{E}_{n}$ and $\mu_{n,0}\colon\mathcal{E}_{n}\otimes_{A}A\hookrightarrow\mathcal{E}_{n}$ be induced by the $A$-bimodule structure on $\mathcal{E}_{n}$ for each $n\in\mathbb{N}$. Then we speak of a weak partial product system. Weak partial product systems on von Neumann algebras have already been used in the study of $E_{0}$-semigroups, where they are called “superproduct systems” (see [Margetts-Srinivasan:E0_II1, Bikram:CAR_flows_III]).

For a partial product system, we impose two more conditions to get a well-behaved theory. Their role is similar to the compact alignment condition for product systems over quasi-lattice orders. The correspondences $\mathcal{E}_{n}$ in a weak partial product systems over $\mathbb{N}$ are much more independent than in an ordinary product system, and so the freeness of the monoid $\mathbb{N}$ no longer helps. This makes compact alignment and Nica covariance relevant already over $\mathbb{N}$. Our Toeplitz algebra is, in fact, an analogue of the Nica–Toeplitz algebra.

Our first goal is to define the Toeplitz $\mathrm{C}^{*}$-algebra of a partial product system. We define partial product systems so that, on the one hand, this $\mathrm{C}^{*}$-algebra has a universal property for suitable representations of the partial product system and, on the other hand, is generated concretely by an analogue of the Fock representation. The definition of a representation has some obvious data and conditions and a non-obvious condition needed to make the Toeplitz $\mathrm{C}^{*}$-algebra well-behaved. We first discuss our definition of a representation. Then we discuss the Fock representation. Only then can we formulate the remaining two conditions on partial product systems. They say simply that the Fock representation exists and is a representation.

Definition 1.1**.**

Let $B$ be a $\mathrm{C}^{*}$-algebra. A weak representation of a weak partial product system $(A,\mathcal{E}_{n},\mu_{n,m})_{n,m\in\mathbb{N}}$ in $B$ consists of linear maps $\omega_{n}\colon\mathcal{E}_{n}\to B$ for all $n\in\mathbb{N}$, such that

1. (1)

$\omega_{n}(x)\cdot\omega_{m}(y)=\omega_{n+m}(\mu_{n,m}(x\otimes y))$ for all $n,m\in\mathbb{N}$, $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$; 2. (2)

$\omega_{n}(x)^{*}\omega_{n}(y)=\omega_{0}(\langle x\,|\,\mathopen{}y\rangle)$ for all $n\in\mathbb{N}$, $x,y\in\mathcal{E}_{n}$;

A weak representation is a representation if, in addition,

1. (3)

$\omega_{n}(\mathcal{E}_{n})^{*}\cdot\omega_{m}(\mathcal{E}_{m})\subseteq\omega_{m-n}(\mathcal{E}_{m-n})\cdot B$ for all $n,m\in\mathbb{N}$ with $m>n>0$; 2. (4)

$\omega_{n}(\mathcal{E}_{n})^{*}\cdot\omega_{m}(\mathcal{E}_{m})\subseteq\omega_{n-m}(\mathcal{E}_{n-m})^{*}\cdot B$ for all $n,m\in\mathbb{N}$ with $0<m<n$.

By convention, $X\cdot Y$ for two subspaces in a $\mathrm{C}^{*}$-algebra $B$ always denotes the closed linear span of the products $x\cdot y$ for $x\in X$, $y\in Y$.

Conditions (1) and (2) for $n=m=0$ hold if and only if $\omega_{0}$ is a ∗-homomorphism. If $n\in\mathbb{N}$, then (1) for $(n,0)$ and $(0,n)$ and (2) for $n$ say that $\omega_{n}$ is a (Toeplitz) representation of the $\mathrm{C}^{*}$-correspondence $\mathcal{E}_{n}$.

The definition of a partial product system uses the Fock representation. This should be a representation in $\mathbb{B}(\mathcal{F})$, where $\mathcal{F}$ is the Hilbert $A$-module direct sum $\bigoplus_{n\in\mathbb{N}}\mathcal{E}_{n}$. For $n,m\in\mathbb{N}$, $x\in\mathcal{E}_{n}$, define

[TABLE]

This map is linear and $\lVert S_{n,m}(x)\rVert\leq\lVert x\rVert$. The first assumption for a partial product system asks $S_{n,m}(x)$ to be adjointable for all $n,m\in\mathbb{N}$, $x\in\mathcal{E}_{n}$. Then the operators $S_{n,m}(x)$ for $m\in\mathbb{N}$ combine to an operator $S_{n}(x)\in\mathbb{B}(\mathcal{F})$ and the maps $S_{n}$ form a weak representation. The second assumption for a partial product system is that they even form a representation, which we call the Fock representation:

Definition 1.2**.**

A partial product system is a weak partial product system for which the Fock representation exists and is a representation.

To understand this condition better, we reformulate (3) and (4) in Definition 1.1 in case $S_{n}(x)$ is adjointable for all $n\in\mathbb{N}$, $x\in\mathcal{E}_{n}$. Then (3) and (4) are equivalent to the first two cases in the following equation:

[TABLE]

here $n,m\in\mathbb{N}$, $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$. The cases $n<m$ and $n>m$ in (1.3) are equivalent to each other by taking adjoints, and the case $n=m$ is condition (2) in Definition 1.1. So a weak partial product system is a partial product system if and only if the operators $S_{n}(x)$ on $\mathcal{F}$ are adjointable for all $n\in\mathbb{N}$, $x\in\mathcal{E}_{n}$ and satisfy

[TABLE]

for all $n,m\in\mathbb{N}$, $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$ with $m>n$. And a weak representation of a partial product system is a representation if and only if $\omega_{n}(x)^{*}\omega_{m}(y)=\omega_{m-n}(S_{n}(x)^{*}y)$ for all $n,m\in\mathbb{N}$, $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$ with $m>n$. The formulation of the extra conditions in Definition 1.1 is inspired by a similar treatment of the Cuntz–Pimsner covariance condition of a proper product system in [Albandik-Meyer:Colimits] and makes sense without the adjointability of $S_{n}(x)$. The reformulation in (1.3) guarantees that there is a universal $\mathrm{C}^{*}$-algebra for representations of our partial product system.

We describe an example of a weak partial product system where (1.4) fails. Its definition uses correspondences based on graphs. A graph with vertex and edge sets $V$ and $E$ gives a $\mathrm{C}^{*}$-correspondence $\mathrm{C}^{*}(E)$ over $\mathrm{C}_{0}(V)$. To define a weak partial product system using graphs, we need a common vertex set $V$, graphs $\Gamma_{n}=(V,E_{n},r_{n},s_{n})$ for all $n\in\mathbb{N}$, and associative, injective multiplication maps $\mu_{n,m}\colon E_{n}\times_{s,r}E_{m}\hookrightarrow E_{n,m}$, where $E_{0}=V$ and $r_{0},s_{0}$ are the identity map, and $\mu_{0,n}$ and $\mu_{n,0}$ are the canonical maps. We study when $\mu_{n,m}$ induces an isometry $\mathrm{C}^{*}(E_{n})\otimes_{\mathrm{C}_{0}(V)}\mathrm{C}^{*}(E_{m})\hookrightarrow\mathrm{C}^{*}(E_{n+m})$ and when these isometries form a partial product system. These conditions are rather restrictive. As it turns out, the category with object set $V$ and arrow set $\bigsqcup_{n\in\mathbb{N}}E_{n}$ must be the path category of an ordinary graph. The only variation is that the grading is not the standard one, that is, elements of $E_{n}$ need not be paths of length $n$.

The following theorem generalises an important feature of Pimsner’s Toeplitz algebras. Our definitions above are arranged so as to make it true.

Theorem 1.5**.**

Let $\mathcal{E}=(A,\mathcal{E}_{n},\mu_{n,m})_{n,m\in\mathbb{N}}$ be a partial product system. Let $(S_{n})_{n\in\mathbb{N}}$ be its Fock representation. The closed linear span of $S_{n}(\mathcal{E}_{n})S_{m}(\mathcal{E}_{m})^{*}$ for $m,n\in\mathbb{N}$ is a $\mathrm{C}^{*}$-subalgebra $\mathcal{T}$ of $\mathbb{B}(\mathcal{F})$, and the maps $S_{n}$ form a representation $\bar{\omega}_{n}$ of $\mathcal{E}$ in $\mathcal{T}$. This representation in $\mathcal{T}$ is universal: for any representation $(\omega_{n}\colon\mathcal{E}_{n}\to B)_{n\in\mathbb{N}}$ of $\mathcal{E}$ there is a unique ∗-homomorphism $\varrho\colon\mathcal{T}\to B$ with $\omega_{n}=\varrho\circ\bar{\omega}_{n}$ for all $n\in\mathbb{N}$.

We describe a gauge action of the circle group $\mathbb{T}$ on $\mathcal{T}$ and prove a gauge-equivariant uniqueness theorem for $\mathcal{T}$. This allows us to prove Theorem 1.5. We also describe the fixed-point subalgebra of the gauge action explicitly as an inductive limit $\mathrm{C}^{*}$-algebra. For $m,n\in\mathbb{N}$, there is a unique linear map

[TABLE]

with

[TABLE]

for all $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$. These maps are injective and their images are linearly independent subspaces of $\mathcal{T}$. Their direct sum over all $m,n\in\mathbb{N}$ is a dense ∗-subalgebra in $\mathcal{T}$ because of (1.3).

We classify gauge-invariant ideals $H\mathrel{\triangleleft}\mathcal{T}$ in the Toeplitz $\mathrm{C}^{*}$-algebra by two ideals in $A$, namely, the kernel $A\cap H$ and the covariance ideal, consisting of all $a\in A$ that are equal modulo $H$ to an element in the closed linear span of $\mathbb{K}(\mathcal{E}_{i})$ for $i\geq 1$. Theorem 5.3 describes $H$ through its kernel and covariance ideal. The kernel $I$ is always an invariant ideal, and any invariant ideal may occur. We do not know, in general, which covariance ideals are possible. We also define what it means for a representation to be covariant on an ideal. If an ideal $J$ is the covariance ideal of some representation, then there is a universal $\mathrm{C}^{*}$-algebra $\mathcal{O}(\mathcal{E},J)$ for representations of $\mathcal{E}$ that are covariant on $J$. We characterise when the canonical representation of $\mathcal{E}$ in $\mathcal{O}(\mathcal{E},J)$ is faithful: this happens if and only if $J\subseteq K^{\bot}$ for $K\mathrel{\vcentcolon=}\bigcap_{n=1}^{\infty}\ker(\vartheta_{0}^{n})$, where $\vartheta_{0}^{n}\colon A\to\mathbb{B}(\mathcal{E}_{n})$ is the left action in the correspondence $\mathcal{E}_{n}$. This allows us to define an analogue of Katsura’s $\mathrm{C}^{*}$-algebra for partial product systems. We also define an analogue of Pimsner’s $\mathrm{C}^{*}$-algebra as the quotient of $\mathcal{T}$ by the closure of the finite block matrices in the Fock representation. For global product systems, we show that the Katsura and Pimsner $\mathrm{C}^{*}$-algebras defined here agree with those previously constructed. We show that the Katsura $\mathrm{C}^{*}$-algebra of a Fell bundle over $\mathbb{Z}$, restricted to a partial product system over $\mathbb{N}$, is the section $\mathrm{C}^{*}$-algebra of the Fell bundle. Using this, we characterise when a partial product system over $\mathbb{N}$ is the restriction of a Fell bundle over $\mathbb{Z}$. For the partial product systems built from graphs, we show that the Katsura algebra is the graph $\mathrm{C}^{*}$-algebra.

2. Weak partial product systems and representations

In this section, we recall some basic notions and study general properties of weak partial product systems and their weak representations. We examine when a weak Fock representation exists, that is, when the operators $S_{n}(x)$ on the Fock module mentioned in the introduction are adjointable. We illustrate our theory with $\mathrm{C}^{*}$-correspondences built from graphs.

Let $A$, $B$, $C$ be $\mathrm{C}^{*}$-algebras. An $A,B$-correspondence is a right Hilbert module $\mathcal{E}$ over $B$ with a ∗-homomorphism $\vartheta\colon A\to\mathbb{B}(\mathcal{E})$ satisfying $\langle\vartheta(a)x\,|\,\mathopen{}y\rangle_{B}=\langle x\,|\,\mathopen{}\vartheta(a)^{*}y\rangle_{B}$ for all $a\in A$, $x,y\in\mathcal{E}$. We often write $ax$ instead of $\vartheta(a)(x)$. We do not require the left action of $A$ on $\mathcal{E}$ to be nondegenerate, and we allow representations of $\mathrm{C}^{*}$-algebras to be degenerate throughout this article.

Let $\mathcal{E}$ and $\mathcal{F}$ be an $A,B$- and a $B,C$-correspondence. We equip the algebraic tensor product $\mathcal{E}\odot\mathcal{F}$ with the obvious $A,C$-bimodule structure $a(x\otimes y)c=ax\otimes yc$ for $a\in A$, $c\in C$, $x\in\mathcal{E}$, $y\in\mathcal{F}$, and with the $C$-valued inner product

[TABLE]

for $x_{1},x_{2}\in\mathcal{E}$ and $y_{1},y_{2}\in\mathcal{F}$. The Hausdorff completion of $\mathcal{E}\odot\mathcal{F}$ for this inner product is an $A,C$-correspondence denoted by $\mathcal{E}\otimes_{B}\mathcal{F}$ (see [Lance:Hilbert_modules]). The identity correspondence $\mathcal{A}$ on $A$ for a $\mathrm{C}^{*}$-algebra $A$ is $A$ viewed as an $A,A$-correspondence using the obvious bimodule structure and the inner product $\langle a\,|\,\mathopen{}b\rangle_{A}\mathrel{\vcentcolon=}a^{*}b$.

Lemma 2.2**.**

Let ${}_{A}\mathcal{E}_{B}$, ${}_{B}\mathcal{F}_{C}$ and ${}_{C}\mathcal{G}_{D}$ be $\mathrm{C}^{*}$-correspondences between the indicated $\mathrm{C}^{*}$-algebras $A,B,C,D$. There are canonical isomorphisms of correspondences

[TABLE]

We usually omit parentheses in tensor products and the associator isomorphism in Lemma 2.2 to reduce the size of our diagrams. They are canonical enough that this cannot cause confusion.

Definition 2.3**.**

A weak partial product system over $\mathbb{N}$ consists of

• •

a $\mathrm{C}^{*}$-algebra $A$,

• •

$A,A$-correspondences $\mathcal{E}_{n}$ for $n\in\mathbb{N}_{\geq 1}$,

• •

isometric bimodule maps $\mu_{n,m}\colon\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}\hookrightarrow\mathcal{E}_{n+m}$ for $n,m\in\mathbb{N}_{\geq 1}$;

such that the following diagrams commute for all $n,m,l\in\mathbb{N}_{\geq 1}$ (“associativity”):

[TABLE]

(Being isometric means that $\langle\iota(x)\,|\,\mathopen{}\iota(y)\rangle=\langle x\,|\,\mathopen{}y\rangle$ for all $x,y\in\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}$.) Let $\mathcal{E}_{0}\mathrel{\vcentcolon=}A$ and let $\mu_{0,m}\colon\mathcal{A}\otimes_{A}\mathcal{E}_{m}\hookrightarrow\mathcal{E}_{m}$ and $\mu_{m,0}\colon\mathcal{E}_{m}\otimes_{A}\mathcal{A}\hookrightarrow\mathcal{E}_{m}$ be the canonical isometries from Lemma 2.2. Then the diagram (2.4) commute for all $n,m,l\in\mathbb{N}$.

Let $(A,\mathcal{E}_{n},\mu_{n,m})_{n,m\in\mathbb{N}}$ be a weak partial product system. Let $(\omega_{n})_{n\in\mathbb{N}}$ be a weak representation of it in a $\mathrm{C}^{*}$-algebra $B$ as in Definition 1.1. That is, $\omega_{n}\colon\mathcal{E}_{n}\to B$ for $n\in\mathbb{N}$ are linear maps satisfying the conditions (1) and (2) in Definition 1.1. Let $m,n\in\mathbb{N}$. By definition, $\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})$ is the closed linear span in $\mathbb{B}(\mathcal{E}_{m},\mathcal{E}_{n})$ of $\lvert x\rangle\langle y\rvert$ for $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$, where $(\lvert x\rangle\langle y\rvert)(z)\mathrel{\vcentcolon=}x\langle y\,|\,\mathopen{}z\rangle_{A}$ for all $z\in\mathcal{E}_{m}$. There is a unique map $\Theta_{m,n}\colon\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})\to B$ with

[TABLE]

for all $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$; this follows from [Pimsner:Generalizing_Cuntz-Krieger]*Lemma 3.2 applied to the representation $\omega_{m}\oplus\omega_{n}\colon\mathcal{E}_{m}\oplus\mathcal{E}_{n}\to\mathbb{M}_{2}(B)$ of $\mathcal{E}_{m}\oplus\mathcal{E}_{n}$, by viewing $\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})$ as an off-diagonal corner in $\mathbb{K}(\mathcal{E}_{m}\oplus\mathcal{E}_{n})$. These maps are compatible with the multiplication maps and adjoints, that is,

[TABLE]

for all $S\in\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})$, $T\in\mathbb{K}(\mathcal{E}_{l},\mathcal{E}_{m})$; this follows from the case of rank-one operators, which in turn follows easily from the conditions in Definition 1.1:

[TABLE]

In particular,

[TABLE]

for all $x\in\mathcal{E}_{n}\cong\mathbb{K}(A,\mathcal{E}_{n})$. So (2.6) implies

[TABLE]

for all $m,n\in\mathbb{N}$, $T\in\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})$, $x\in\mathcal{E}_{m}$.

The maps $\Theta_{m,n}$ map $\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})$ into the $\mathrm{C}^{*}$-subalgebra of $B$ that is generated by $\omega_{n}(\mathcal{E}_{n})$. The closed linear span of $\Theta_{m,n}(\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n}))$ for $m,n\in\mathbb{N}$ need not be an algebra. We will impose more relations to arrange for this later.

Definition 2.8**.**

An ideal $J\mathrel{\triangleleft}A$ is invariant with respect to a weak partial product system $\mathcal{E}=(A,\mathcal{E}_{n},\mu_{n,m})_{n,m\in\mathbb{N}}$ if $J\cdot\mathcal{E}_{n}\subseteq\mathcal{E}_{n}\cdot J$ for all $n\in\mathbb{N}$.

Lemma 2.9**.**

Let $(\omega_{n})_{n\in\mathbb{N}}$ be a weak representation of a weak partial product system $\mathcal{E}$ in a $\mathrm{C}^{*}$-algebra $B$. The ideal $I\mathrel{\vcentcolon=}\ker\omega_{0}$ is invariant and

[TABLE]

Proof.

Let $x\in\mathcal{E}_{n}$. We have $\omega_{n}(x)=0$ if and only if $0=\omega_{n}(x)^{*}\omega_{n}(x)=\omega_{0}(\langle x\,|\,\mathopen{}x\rangle)$, if and only if $\langle x\,|\,\mathopen{}x\rangle\in I$. The latter is equivalent to $x\in\mathcal{E}_{n}\cdot I$. So $\ker\omega_{n}=\mathcal{E}_{n}\cdot I$. This implies that $I$ is invariant because $I\cdot\mathcal{E}_{n}\subseteq\ker\omega_{n}$. Let $T\in\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})$. Then $T\in\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n}\cdot I)$ if and only if $T(x)\in\mathcal{E}_{n}\cdot I$ for all $x\in\mathcal{E}_{m}$. By the first part, this is equivalent to $\omega_{n}(T(x))=\Theta_{m,n}(T)\omega_{m}(x)=0$ for all $x\in\mathcal{E}_{m}$. This follows if $\Theta_{m,n}(T)=0$. Conversely, if $\Theta_{m,n}(T)\omega_{m}(x)=0$ for all $x\in\mathcal{E}_{m}$, then $\Theta_{m,n}(T)\cdot\Theta_{n,m}(S)=0$ for all $S\in\mathbb{K}(\mathcal{E}_{n},\mathcal{E}_{m})$. Taking $S=T^{*}$, this implies $\Theta_{m,n}(T)=0$ because $\Theta_{n,m}(T^{*})=\Theta_{m,n}(T)^{*}$. ∎

Next we seek an analogue of the Fock representation. We want this to exist because it is used by Pimsner [Pimsner:Generalizing_Cuntz-Krieger] to define the Toeplitz $\mathrm{C}^{*}$-algebra. The Fock representation should be a (weak) representation on the Hilbert $A$-module $\mathcal{F}\mathrel{\vcentcolon=}\bigoplus_{n=0}^{\infty}\mathcal{E}_{n}$, which we call the Fock module of $\mathcal{E}$. Fix $n\in\mathbb{N}$ and $x\in\mathcal{E}_{n}$. In the Fock representation, $x$ should act on the summand $\mathcal{E}_{m}$ by the operator

[TABLE]

More precisely, $S_{n,m}(x)$ is the composite of the isometry $\mu_{n,m}$ with the creation operator $\mathcal{E}_{m}\to\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}$, $y\mapsto x\otimes y$, which is adjointable. So the operator above is a well defined bounded linear map. It is always adjointable for $n=0$, but not for $n>0$. Sufficient conditions for this are the following:

1. (1)

if the isometries $\mu_{n,m}$ are adjointable; 2. (2)

if the correspondence $\mathcal{E}_{n}$ is proper, that is, $A$ acts by compact operators on $\mathcal{E}_{n}$: then the creation operator $\mathcal{E}_{m}\to\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}$ is compact, and then so is $S_{n,m}(x)$ because $\mathbb{K}(\mathcal{F})\subseteq\mathbb{K}(\mathcal{E})$ if $\mathcal{F}\subseteq\mathcal{E}$ is a Hilbert submodule in a Hilbert module; 3. (3)

if $(\mathcal{E}_{n})_{n\in\mathbb{N}}$ comes from a Fell bundle over $\mathbb{Z}$: then the left multiplication map $\mathcal{E}_{n+m}\to\mathcal{E}_{m}$, $y\mapsto x^{*}\cdot y$, for $x^{*}\in\mathcal{E}_{-n}$ is adjoint to $S_{n,m}(x)$.

Condition (1) contains product systems in the usual sense, where each $\mu_{n,m}$ is unitary and hence adjointable.

The adjointability of the operators $S_{n,m}(x)$ is one of the requirements for a partial product system. In other words, we require that for all $x\in\mathcal{E}_{n}$, $t\in\mathcal{E}_{n+m}$, there is $z\in\mathcal{E}_{m}$, necessarily unique, with $\langle t\,|\,\mathopen{}x\cdot y\rangle_{A}=\langle z\,|\,\mathopen{}y\rangle_{A}$ for all $y\in\mathcal{E}_{m}$. If the maps $S_{n,m}(x)$ are adjointable, then so is the creation operator $S_{n}(x)\mathrel{\vcentcolon=}\sum_{m\in\mathbb{N}}S_{n,m}(x)$ on the Fock module $\mathcal{F}$; we call the adjoint $S_{n}(x)^{*}$ an annihilation operator. It is easy to see that the maps $S_{n}(x)$ form a weak representation of our weak partial product system in $\mathbb{B}(\mathcal{F})$. That is, $S_{n}(x)S_{m}(y)=S_{n+m}(x\cdot y)$ for $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$, $n,m\in\mathbb{N}_{\geq 1}$, and $S_{n}(x)^{*}S_{n}(y)=S_{0}(\langle x\,|\,\mathopen{}y\rangle_{A})$ for $x,y\in\mathcal{E}_{n}$, $n,m\in\mathbb{N}_{\geq 1}$, $n\in\mathbb{N}_{\geq 1}$.

2.1. Correspondences associated to graphs

A (directed) graph is given by countable discrete sets $V$ and $E$ of vertices and edges and maps $(r,s)\colon E\rightrightarrows V$ sending an edge to its range and source. It yields a $\mathrm{C}_{0}(V),\mathrm{C}_{0}(V)$-correspondence by completing $\mathrm{C_{c}}(E)$ in the $\mathrm{C}_{0}(V)$-valued inner product

[TABLE]

for $x,y\in\mathrm{C_{c}}(E)$ and $v\in V$ or, equivalently,

[TABLE]

the $\mathrm{C}_{0}(V)$-bimodule structure is given by

[TABLE]

for $e\in E$, $a\in\mathrm{C}_{0}(V)$, $x\in\mathrm{C_{c}}(E)$. This is a $\mathrm{C}^{*}$-correspondence, and its Cuntz–Pimsner algebra (as modified by Katsura) is the graph $\mathrm{C}^{*}$-algebra of our graph (see [Raeburn:Graph_algebras]).

Now consider two graphs with the same vertex set $V$, with sets of edges $E_{1}$ and $E_{2}$ and range and source maps $(r_{i},s_{i})\colon E_{i}\rightrightarrows V$ for $i=1,2$. As above, we build two $\mathrm{C}_{0}(V),\mathrm{C}_{0}(V)$-correspondences $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$. The composite correspondence $\mathcal{E}_{1}\otimes_{\mathrm{C}_{0}(V)}\mathcal{E}_{2}$ is associated to the graph with edge set $E\mathrel{\vcentcolon=}E_{1}\times_{s,r}E_{2}$ and $r,s\colon E\rightrightarrows V$ defined by $r(f,g)\mathrel{\vcentcolon=}r_{1}(f)$ and $s(f,g)\mathrel{\vcentcolon=}s_{2}(g)$.

Now let $r_{n},s_{n}\colon E_{n}\rightrightarrows V$ for $n\in\mathbb{N}_{>0}$ be graphs with the same vertex set $V$. Let $\tilde{\mu}_{n,m}\colon E_{n}\times_{s,r}E_{m}\to E_{n+m}$ be injective maps for all $n,m\in\mathbb{N}$, which we write multiplicatively as $x\cdot y\mathrel{\vcentcolon=}\tilde{\mu}_{n,m}(x,y)$ for $x\in E_{n}$, $y\in E_{m}$ with $s_{n}(x)=r_{m}(y)$. Assume that $r_{n+m}(x\cdot y)=r_{n}(x)$ and $s_{n+m}(x\cdot y)=s_{m}(y)$ and that these multiplication maps are associative. Let $E_{0}=V$ and $r_{0}=s_{0}=\mathrm{id}_{V}$ and let the multiplication maps $\tilde{\mu}_{0,m}\colon V\times_{s,r}E_{m}\to E_{m}$ and $\tilde{\mu}_{m,0}\colon E_{m}\times_{s,r}V\to E_{m}$ be the obvious maps $(r(x),x)\mapsto x$, $(x,s(x))\mapsto x$. We are going to build a weak partial product system out of this data. The construction will also show that the assumptions above are necessary to get a weak partial product system.

Let $A\mathrel{\vcentcolon=}\mathrm{C}_{0}(V)$ and let $\mathcal{E}_{n}$ be the $A,A$-correspondence associated to the graph $E_{n}$ as above. The characteristic functions $(\delta_{x})_{x\in E_{n}}$ form a basis in $\mathrm{C_{c}}(E_{n})$, which is dense in $\mathcal{E}_{n}$. So $(\delta_{x}\otimes\delta_{y})_{x\in E_{n},y\in E_{m}}$ is a basis in $\mathrm{C_{c}}(E_{n})\odot\mathrm{C_{c}}(E_{m})$, which maps to a dense subset in $\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}$. For $x\in E_{n}$, $y\in E_{m}$, define

[TABLE]

Lemma 2.10**.**

The map $\mu_{n,m}\colon\mathrm{C_{c}}(E_{n})\odot\mathrm{C_{c}}(E_{m})\to\mathrm{C_{c}}(E_{n+m})$ is isometric for the $\mathrm{C}_{0}(V)$-valued inner product and extends uniquely to an isometric bimodule map $\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}\hookrightarrow\mathcal{E}_{n+m}$.

Proof.

Let $x,x^{\prime}\in E_{n}$, $y,y^{\prime}\in E_{m}$. We write $\delta_{a=b}$ for the Kronecker delta to distinguish it more clearly from our characteristic functions. We declare that $\delta_{x\cdot y}=0$ if $s_{n}(x)\neq r_{m}(y)$, so that $\mu_{n,m}(\delta_{x}\otimes\delta_{y})=\delta_{x\cdot y}$ holds for all $x\in E_{n}$, $y\in E_{m}$. On the one hand, we compute

[TABLE]

On the other hand, we compute

[TABLE]

Since $\tilde{\mu}_{n,m}$ is injective, $x\cdot x^{\prime}=y\cdot y^{\prime}$ if and only if $x=x^{\prime}$ and $y=y^{\prime}$. We assumed $s(x\cdot y)=s(y)$ as well. So both expressions above are equal. This shows that $\mu_{n,m}$ preserves the inner products between the elements $\delta_{x}\otimes\delta_{y}$ and $\delta_{x^{\prime}}\otimes\delta_{y^{\prime}}$ for $(x,y),(x^{\prime},y^{\prime})\in E_{n}\times_{s,r}E_{m}$. Hence it induces an isometry $\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}\hookrightarrow\mathcal{E}_{n+m}$.

The condition $r_{n+m}(x\cdot y)=r_{n}(x)$ for all $(x,y)\in E_{n}\times_{s,r}E_{m}$ implies that $\mu_{n,m}$ is a left module homomorphism. The condition $s_{n+m}(x\cdot y)=s_{m}(y)$ implies that it is a right module homomorphism. ∎

The associativity condition in (2.4) is clearly equivalent to

[TABLE]

for all $x\in E_{n}$, $y\in E_{m}$, $z\in E_{l}$, $n,m,l\in\mathbb{N}_{>0}$. Thus we obtain a weak partial product system when we add this to our requirements. Our choices for $E_{0}$, $\tilde{\mu}_{0,m}$ and $\tilde{\mu}_{m,0}$ say that $\mathcal{E}_{0}=\mathcal{A}$ and that $\mu_{0,m}$ and $\mu_{m,0}$ are the canonical maps $\mathcal{A}\otimes_{A}\mathcal{E}_{m}\hookrightarrow\mathcal{E}_{m}$ and $\mathcal{E}_{m}\otimes_{A}\mathcal{A}\xrightarrow{\sim}\mathcal{E}_{m}$ as in Lemma 2.2.

The disjoint union $E\mathrel{\vcentcolon=}\bigsqcup_{n\in\mathbb{N}}E_{n}$ is a category with object set $V$, using the multiplication maps $\tilde{\mu}_{n,m}$ for $n,m\in\mathbb{N}$. The decomposition of $E$ as a disjoint union may be encoded by the functor from $E$ to the monoid $(\mathbb{N},+)$ which maps elements of $E_{n}$ to $n$. We want to describe a representation $(\omega_{n})_{n\in\mathbb{N}}$ of the weak partial product system above through representations of this category. The representation $\omega_{n}$ of $\mathcal{E}_{n}$ is given by the operators $T_{x}\mathrel{\vcentcolon=}\omega_{n}(\delta_{x})$ for $x\in E_{n}$. Since $\omega_{0}$ is a representation of $\mathrm{C}_{0}(V)$, the operators $T_{v}$ for $v\in E_{0}=V$ are orthogonal projections. Conditions (1) and (2) in Definition 1.1 say, first, that $T_{x}T_{y}$ is $T_{x\cdot y}$ if $s(x)=r(y)$, and [math] otherwise and, secondly, that $T_{x}^{*}T_{y}=\delta_{x,y}T_{s(y)}$ if $x,y\in E_{n}$ for the same $n\in\mathbb{N}$. So each $T_{x}$ is an isometry from $T_{s(x)}B\subseteq B$ into $T_{r(x)}B\subseteq B$, and the ranges of these isometries for $x\in E_{n}$ with fixed $n\in\mathbb{N}$ are orthogonal. If $V$ has only one element, then our category $E$ becomes a monoid, and the map $x\mapsto T_{x}$ is a representation of this monoid by isometries with the extra property that the isometries $T_{x}$ for $x\in E_{n}$ with fixed $n\in\mathbb{N}$ have orthogonal ranges.

We now examine the Fock representation. The Fock module $\mathcal{F}$ is the Hilbert $\mathrm{C}_{0}(V)$-module with basis $(\delta_{x})_{x\in E}$ with $E=\bigsqcup E_{n}$ as above, and with the inner product $\langle\delta_{x}\,|\,\mathopen{}\delta_{y}\rangle=\delta_{x=y}\delta_{s(x)}$. The creation operator for $x\in E$ acts on $\mathcal{F}$ by $S_{\delta_{x}}(\delta_{y})\mathrel{\vcentcolon=}\delta_{x\cdot y}$ for $x,y\in E$. Since the multiplication map on $E_{n}\times_{s,r}E_{m}$ is injective for each $n,m\in\mathbb{N}$ by assumption, the map $S_{\delta_{x}}$ is a partial isometry with domain spanned by $\delta_{y}$ with $r(y)=s(x)$ and image spanned by $\delta_{x\cdot y}$ for all such $y$. This image is complementable, the complement being spanned by those $\delta_{y}$ with $y\in E\setminus(x\cdot E)$. So $S_{\delta_{x}}$ has the adjoint

[TABLE]

Since the adjointable operators form a Banach space, it follows that $S_{\xi}$ is adjointable for all $\xi\in\mathcal{E}_{n}$. Hence the Fock representation exists as a weak representation.

3. Representations and partial product systems

In this section, we restrict attention to weak partial product systems for which the weak Fock representation exists. We show that the extra conditions for a representation, (3) and (4) in Definition 1.1, are equivalent to (1.3). We show an example as in Section 2.1 for which these conditions fail for the Fock representation. We define partial product systems by requiring that the Fock representation be defined and be a representation. We study the extra conditions needed for representations for global product systems and Fell bundles. Finally, we relate our notion of representation and partial product system to Nica covariance and compact alignment for product systems over quasi-lattice orders.

Proposition 3.1**.**

Let $\mathcal{E}=(A,\mathcal{E}_{n},\mu_{n,m})_{n,m\in\mathbb{N}}$ be a weak partial product system for which the weak Fock representation exists, that is, the creation operators on its Fock module are adjointable. Let $(\omega_{n})_{n\in\mathbb{N}}$ be a weak representation of $\mathcal{E}$ in a $\mathrm{C}^{*}$-algebra $B$. Define $\Theta_{m,n}\colon\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})\to B$ as in (2.5). The following are equivalent:

1. (1)

$\omega_{n}(x)^{*}\omega_{m}(y)=\omega_{m-n}(S_{n}(x)^{*}y)$* for all $n,m\in\mathbb{N}_{\geq 1}$ with $m>n$ and $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$;* 2. (2)

$\omega_{n}(x)^{*}\omega_{m}(y)=\omega_{n-m}(S_{m}(y)^{*}x)^{*}$* for all $n,m\in\mathbb{N}_{\geq 1}$ with $n>m$ and $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$;* 3. (3)

$\omega_{n}(\mathcal{E}_{n})^{*}\cdot\omega_{m}(\mathcal{E}_{m})\subseteq\omega_{m-n}(\mathcal{E}_{m-n})$* for all $n,m\in\mathbb{N}_{>0}$ with $m>n$;* 4. (4)

$\omega_{n}(\mathcal{E}_{n})^{*}\cdot\omega_{m}(\mathcal{E}_{m})\subseteq\omega_{n-m}(\mathcal{E}_{n-m})^{*}$* for all $n,m\in\mathbb{N}_{>0}$ with $m<n$;* 5. (5)

$\omega_{n}(\mathcal{E}_{n})^{*}\cdot\omega_{m}(\mathcal{E}_{m})\cdot B\subseteq\omega_{m-n}(\mathcal{E}_{m-n})\cdot B$* for all $n,m\in\mathbb{N}$ with $m>n$;* 6. (6)

$\omega_{n}(\mathcal{E}_{n})^{*}\cdot\omega_{m}(\mathcal{E}_{m})\cdot B\subseteq\omega_{n-m}(\mathcal{E}_{n-m})^{*}\cdot B$* for all $n,m\in\mathbb{N}_{>0}$ with $m<n$.* 7. (7)

If $m,n,p,q\in\mathbb{N}$, then

[TABLE]

These equivalent conditions characterise when $(\omega_{n})_{n\in\mathbb{N}}$ is a representation.

Proof.

The condition in (2) is the adjoint of the condition in (1). So these two conditions are equivalent. Similarly, (3) and (4) are equivalent. The implications

[TABLE]

are trivial. We are going to show that (5) implies (1). An analogous argument shows that (6) implies (2). This will complete the proof that conditions (1)–(6) are equivalent. A representation is a weak representation that also satisfies the conditions (5) and (6). Hence each of our equivalent conditions characterises representations among weak representations.

Let $z\in\mathcal{E}_{m-n}$. Then

[TABLE]

Letting

[TABLE]

this becomes $\omega_{m-n}(z)^{*}\cdot X=0$ for all $z\in\mathcal{E}_{m-n}$ or $X^{*}\cdot\omega_{m-n}(\mathcal{E}_{m-n})\cdot B=0$. Condition (5) implies $X\cdot B\subseteq\omega_{m-n}(\mathcal{E}_{m-n})B$. Thus $X^{*}\cdot X\cdot B=0$. Hence $X=0$.

Condition (7) contains (3) and (4) as special cases because

[TABLE]

It remains to prove that (3) implies (7). By definition,

[TABLE]

Hence

[TABLE]

Now we apply (3) to the two factors in the middle. If $m>q$, then

[TABLE]

both for $q>0$ and $q=0$. If $m<q$, then

[TABLE]

both for $m>0$ and $m=0$. The case $m=q$ also works. Thus (3) implies (7). ∎

Equation (2.6) and condition (7) in Proposition 3.1 imply that

[TABLE]

is a ∗-subalgebra for any representation. For $n=m$ and $p=q$, the condition (7) in Proposition 3.1 says that

[TABLE]

here we abbreviated $\Theta_{n}\mathrel{\vcentcolon=}\Theta_{n,m}$. So $\sum_{n=0}^{N}\Theta_{n}(\mathbb{K}(\mathcal{E}_{n}))$ is a ∗-subalgebra as well for all $N\in\mathbb{N}\cup\{\infty\}$.

The conditions (5) and (6) concern inclusions between certain right ideals (or Hilbert submodules) of $B$. Hence they are similar to nondegeneracy conditions. Thus Proposition 3.1 is similar in spirit to [Albandik-Meyer:Product]*Proposition 2.5.

A partial product system is a weak partial product system for which the weak Fock representation exists and is a representation, that is, satisfies the equivalent conditions in Proposition 3.1. These assumptions will be used in the next section to define and study the Toeplitz algebra of a partial product system.

We now examine the difference between weak representations and representations for several classes of weak partial product systems. First, we examine product systems, then restrictions of Fell bundles over $\mathbb{Z}$. Finally, we study examples coming from graphs as in Section 2.1.

A product system is a weak partial product system $(A,\mathcal{E}_{n},\mu_{n,m})_{n,m\in\mathbb{N}}$ where the multiplication maps $\mu_{n,m}\colon\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}\hookrightarrow\mathcal{E}_{n+m}$ for $n,m\in\mathbb{N}_{>0}$ are unitary.

Proposition 3.3**.**

Any weak representation of a product system is a representation. Product systems are partial product systems.

Proof.

Let $(A,\mathcal{E}_{n},\mu_{n,m})_{n,m\in\mathbb{N}}$ be a product system and let $(\omega_{n})_{n\in\mathbb{N}}$ be a weak representation of it in a $\mathrm{C}^{*}$-algebra $B$. Let $n,k\in\mathbb{N}_{>0}$. Then $\omega_{n}(\mathcal{E}_{n})\cdot\omega_{k}(\mathcal{E}_{k})=\omega_{n+k}(\mathcal{E}_{n+k})$, that is, the closed linear span of $\omega_{n}(\xi)\cdot\omega_{k}(\eta)$ for $\xi\in\mathcal{E}_{n}$, $\eta\in\mathcal{E}_{k}$ is dense in $\omega_{n+k}(\mathcal{E}_{n+k})$. If $m>n$, write $m=n+k$. Then

[TABLE]

So any weak representation satisfies the condition (1) in Proposition 3.1. The Fock representation exists as a weak representation because the maps $\mu_{n,m}$ for $n>0$ are adjointable. It is a representation by the statement already proved. That is, our product system is a partial product system. ∎

A Fell bundle over the group $\mathbb{Z}$ is given by Banach spaces $B_{n}$ for $n\in\mathbb{Z}$ with multiplication maps $B_{n}\times B_{m}\to B_{n+m}$ and involutions $B_{n}\to B_{-n}$ with certain properties (see [Exel:Partial_dynamical]*Definition 16.1). These properties are equivalent to the existence of injective maps $B_{n}\hookrightarrow B$ for some $\mathrm{C}^{*}$-algebra $B$ such that the multiplication maps and involutions in $(B_{n})_{n\in\mathbb{N}}$ are the restrictions of the multiplication and involution in $B$. In particular, $B_{0}$ is a $\mathrm{C}^{*}$-algebra. Each $B_{n}$ becomes a Hilbert $B_{0}$-bimodule by $\langle\!\langle b\,|\,\mathopen{}c\rangle\!\rangle\mathrel{\vcentcolon=}bc^{*}$ and $\langle b\,|\,\mathopen{}c\rangle\mathrel{\vcentcolon=}b^{*}c$ for all $b,c\in B_{n}$. The Fell bundle multiplication maps $B_{n}\times B_{m}\to B_{n+m}$ induce isometries $\mu_{n,m}\colon B_{n}\otimes_{B_{0}}B_{m}\hookrightarrow B_{n+m}$ of $B_{0},B_{0}$-correspondences for all $n,m\in\mathbb{N}$.

Proposition 3.4**.**

Let $\mathcal{B}=(B_{n})_{n\in\mathbb{Z}}$ be a Fell bundle. Its restriction to $\mathbb{N}$ is a partial product system; here each $B_{n}$ is viewed as a $B_{0},B_{0}$-correspondence, even a Hilbert $B_{0}$-bimodule, as above.

Proof.

The creation operator $S_{n}(x)\colon B_{m}\to B_{n+m}$, $y\mapsto x\cdot y$, for $x\in B_{n}$ is adjointable with adjoint $S_{n}(x)^{*}(y)=x^{*}\cdot y$. So

[TABLE]

for $n,m,k\in\mathbb{N}$ with $n\leq m$ and $x\in B_{n}$, $y\in B_{m}$, $z\in B_{k}$. Thus the Fock representation $(S_{n})_{n\in\mathbb{N}}$ satisfies condition (1) in Proposition 3.1. So it is a representation and we have got a partial product system over $\mathbb{N}$. ∎

Definition 3.5**.**

Let $D$ be a $\mathrm{C}^{*}$-algebra. A representation of a Fell bundle $\mathcal{B}=(B_{n})_{n\in\mathbb{Z}}$ in $D$ consists of linear maps $\omega_{n}\colon B_{n}\to D$ for $n\in\mathbb{Z}$ such that

1. (1)

$\omega_{0}\colon B_{1}\to D$ is a ∗-homomorphism; 2. (2)

$\omega_{n}(b)\omega_{m}(c)=\omega_{n+m}(bc)$ for all $n,m\in\mathbb{Z}$, $b\in B_{n}$, $c\in B_{m}$; 3. (3)

$\omega_{-n}(b^{*})=\omega_{n}(b)^{*}$ for all $n\in\mathbb{Z}$, $b\in B_{n}$.

It is immediate from the definitions that the restriction of a representation of a Fell bundle over $\mathbb{Z}$ to $\mathbb{N}$ is a representation of the partial product system over $\mathbb{N}$. The Fock representation, however, is never the restriction of a Fell bundle representation. So a Fell bundle over $\mathbb{Z}$ has strictly fewer representations than its partial product system restriction to $\mathbb{N}$.

When is a (weak) partial product system over $\mathbb{N}$ the restriction of a Fell bundle over $\mathbb{Z}$? We shall answer this question in Theorem 5.16. In this section, we only study some easy aspects of this question. A necessary condition is that each $\mathcal{E}_{n}$ be a Hilbert $A$-bimodule. The following example show that this is not yet sufficient, even if we also assume $\mathcal{E}$ to be a partial product system.

Example 3.6*.*

Let $A=\mathbb{C}\oplus\mathbb{C}$ and let $\varphi$ be the partial isomorphism on $A$ that maps the first summand identically onto the second summand. This partial isomorphism corresponds to the Hilbert $A$-bimodule $\mathcal{E}_{\varphi}=\mathbb{C}$ with the $A$-bimodule structure $(a_{1},a_{2})\cdot x\cdot(b_{1},b_{2})\mathrel{\vcentcolon=}a_{2}\cdot x\cdot b_{1}$ and the left and right inner products $\langle\!\langle x\,|\,\mathopen{}y\rangle\!\rangle\mathrel{\vcentcolon=}(0,x\overline{y})$, $\langle x\,|\,\mathopen{}y\rangle\mathrel{\vcentcolon=}(\overline{x}y,0)$ for $a_{1},a_{2},x,y,b_{1},b_{2}\in\mathbb{C}$. Since the range and source ideals of $\mathcal{E}_{\varphi}$ are orthogonal, $\mathcal{E}_{\varphi}\otimes_{A}\mathcal{E}_{\varphi}\cong 0$. Let $\mathcal{E}_{n}\mathrel{\vcentcolon=}\mathcal{E}_{\varphi}$ for all $n\in\mathbb{N}_{\geq 1}$ and let $\mu_{n,m}\colon\mathcal{E}_{n}\otimes_{A}\mathcal{E}_{m}\cong 0\hookrightarrow\mathcal{E}_{n+m}$ be the zero map for all $n,m\in\mathbb{N}_{\geq 1}$. We claim that this is a partial product system over $\mathbb{N}$. All its fibres are Hilbert $A$-bimodules. The maps $\mu_{n,m}$ are isometric and associative. The creation operator $S_{n}(x)$ for $x\in\mathcal{E}_{n}$ vanishes on $\mathcal{E}_{m}$ for $m\in\mathbb{N}_{\geq 1}$ and maps $(b_{1},b_{2})\in\mathcal{E}_{0}=A$ to $x\cdot b_{1}\in\mathcal{E}_{n}$. Hence $S_{n}(x)$ is adjointable, and its adjoint vanishes on $\mathcal{E}_{m}$ for all $m\neq n$. Thus the Fock representation satisfies condition (1) in Proposition 3.1. So we have a partial product system of Hilbert bimodules. It cannot come from a Fell bundle over $\mathbb{Z}$, however. A Fell bundle also has injective multiplication maps

[TABLE]

We have $\mathcal{E}_{n}^{*}\otimes_{A}\mathcal{E}_{m}=\mathcal{E}_{\varphi}^{*}\otimes_{A}\mathcal{E}_{\varphi}$, which is the Hilbert bimodule that corresponds to the identity map on the first summand in $A$. Since there is no non-zero map from this to $\mathcal{E}_{\varphi}$, there is no multiplication map $\mathcal{E}_{n}^{*}\otimes_{A}\mathcal{E}_{m}\hookrightarrow\mathcal{E}_{m-n}$ for $m>n$.

Remark 3.7*.*

Any product system of Hilbert bimodules is the restriction of a Fell bundle over $\mathbb{Z}$. Indeed, a product system is determined by the correspondence $\mathcal{E}_{1}$, and so a product system of Hilbert bimodules is given by a single Hilbert bimodule. The Fell bundle generated by it is described in [Abadie-Eilers-Exel:Morita_bimodules]. This does not yet give all Fell bundles over $\mathbb{Z}$, however: we only get those Fell bundles that are semi-saturated, that is, have surjective multiplication maps $\mu_{n,m}$ for all $n,m\geq 0$.

There is a weak partial product system of the type introduced in Section 2.1 for which the weak Fock representation exists but is not a representation:

Example 3.8*.*

Let

[TABLE]

and $E_{n}=\emptyset$ for $n\geq 5$ with $a\colon v_{1}\to v_{3}$, $b\colon v_{0}\to v_{1}$, $c\colon v_{2}\to v_{3}$, $d\colon v_{0}\to v_{2}$, and $e=a\cdot b=c\cdot d$. This determines the range and source maps $r_{n},s_{n}\colon E_{n}\rightrightarrows V$ and the multiplication maps $\tilde{\mu}_{n,m}\colon E_{n}\times_{s_{n},r_{m}}E_{m}\hookrightarrow E_{n+m}$ for all $n,m\in\mathbb{N}$. The category $\bigsqcup_{n\in\mathbb{N}}E_{n}$ is the one that describes commutative squares:

[TABLE]

The resulting $\mathrm{C}^{*}$-algebra is $A=\mathbb{C}[V]=\mathbb{C}^{4}$. The Fock module over $A$ is the $\mathbb{C}$-vector space with basis

[TABLE]

with the $A$-bimodule structure given by the range and source maps and with $\langle x\,|\,\mathopen{}y\rangle\mathrel{\vcentcolon=}\delta_{x=y}s(x)$ for all $x,y\in E$. The Fock representation of the weak partial product system is given on the basis vectors by the following table:

[TABLE]

So $S_{1}(\mathcal{E}_{1})\mathcal{F}=\mathbb{C}[a,e]$, $S_{2}(\mathcal{E}_{2})\mathcal{F}=\mathbb{C}[c,d,e]$, and $S_{1}(\mathcal{E}_{1})^{*}(S_{2}(\mathcal{E}_{2})\mathcal{F})=\mathbb{C}[b]$. This is not contained in $S_{1}(\mathcal{E}_{1})\mathcal{F}$.

Proposition 3.9**.**

A weak partial product system of the form built in Section 2.1 is a partial product system if and only if $E=\bigsqcup_{n\in\mathbb{N}}E_{n}$ is the path category of some directed graph $\Gamma$, but where generators in $\Gamma$ may have degrees different from $1$.

Proof.

We have already seen that the weak Fock representation exists, that is, the operators $S_{n}(\xi)$ for $\xi\in\mathcal{E}_{n}$ are adjointable. When is it a representation? This means that $S_{n}(x)^{*}S_{m}(y)z\in S_{m-n}(\mathcal{E}_{m-n})\mathcal{F}$ for $m,n\in\mathbb{N}$ with $m>n$ and all $x\in E_{n}$, $y\in E_{m}$, $z\in E_{k}$. By definition, $S_{n}(\delta_{x})^{*}S_{m}(\delta_{y})(\delta_{z})=S_{n}(\delta_{x})^{*}(\delta_{y\cdot z})$ is equal to $\langle\delta_{x}\,|\,\mathopen{}\delta_{s}\rangle\delta_{t}=\delta_{x=s}\delta_{t}$ if $y\cdot z=s\cdot t$ for some $s\in E_{n}$ and $t\in E_{m+k-n}$, and [math] otherwise. So $S_{n}(\delta_{x})^{*}S_{m}(\delta_{y})(\delta_{z})=\delta_{t}$ if $y\cdot z=x\cdot t$ and [math] otherwise. We need this to belong to $S_{m-n}(\mathcal{E}_{m-n})\mathcal{F}$, that is, to be either [math] or of the form $\delta_{a\cdot b}$ for some $a\in E_{m-n}$, $b\in E_{k}$. Thus the weak Fock representation is a representation if and only if $y\cdot z=x\cdot t$ for $x\in E_{n}$, $y\in E_{m}$, $z\in E_{k}$, $t\in E_{m+k-n}$ implies that $t=a\cdot b$ for some $a\in E_{m-n}$, $b\in E_{k}$. Since we already assumed that the multiplication map $E_{n}\times_{s,r}E_{k}\to E_{n+k}$ is injective, the equation $y\cdot z=(x\cdot a)\cdot b$ implies $z=b$ and $y=x\cdot a$. That is, the equation $y\cdot z=x\cdot t$ has only the trivial solutions with $y=x\cdot a$ and $t=a\cdot z$ in $E$.

We call an element $x\in\bigsqcup_{n=1}^{\infty}E$ irreducible if the only product decompositions $x=a\cdot b$ with $a,b\in E$ are $x=1_{r(x)}\cdot x$ and $x=x\cdot 1_{s(x)}$. The irreducible elements in $E$ with the restriction of the range and source maps form a directed graph $\Gamma$. Since $E$ is a category, any path in the graph $\Gamma$ defines an element in $E$. This defines a functor from the path category of $\Gamma$ to $E$. This functor is the identity on objects. It is surjective on arrows because if an arrow is not irreducible, we may write it as a non-trivial product of two strictly shorter arrows, and decomposing these as long as possible will eventually write our arrow as a product of irreducible arrows.

In a path category, the equation $y\cdot z=x\cdot t$ only has the trivial solutions. Conversely, we claim that the functor from the path category of $\Gamma$ to $E$ is injective if the equation $y\cdot z=x\cdot t$ has only the trivial solutions and the multiplication maps $E_{n}\times_{s,r}E_{m}\to E_{n+m}$ are injective for all $n,m\in\mathbb{N}$. Indeed, assume that two paths $a_{1}\dotsm a_{k}$ and $b_{1}\dotsm b_{l}$ in the graph $\Gamma$ are mapped to the same arrow in $E$. Since $y\cdot z=x\cdot t$ has only trivial solutions and $a_{k}$ and $b_{l}$ are irreducible, we must have $a_{k}=b_{l}$ and $a_{1}\dotsm a_{k-1}=b_{1}\dotsm b_{l-1}$. By induction, we conclude that our two paths are equal. So the functor from the path category of $\Gamma$ to $E$ is an isomorphism of categories if $E$ gives a partial product system. The path category of $\Gamma$ differ from $E$ only through the grading, that is, the functor to $\mathbb{N}$. Whereas all generators in a usual path category have length $1$, they may belong to $E_{n}$ for any $n\in\mathbb{N}$. ∎

Roughly speaking, Proposition 3.9 says that the “partial” analogues of graph $\mathrm{C}^{*}$-algebras are not more general than graph $\mathrm{C}^{*}$-algebras. The only thing that is modified is the gauge action because the edges of the graph $\Gamma$ may belong to $\mathcal{E}_{n}$ and thus have degree $n$ for any $n\in\mathbb{N}$.

3.1. Analogy with Nica covariance

We briefly discuss the analogy between the extra condition for a weak representation to be a representation and Nica covariance for representations of product systems over quasi-lattice orders (see [Fowler:Product_systems]*Definition 5.1). Let $(G,P)$ be a quasi-lattice ordered group; for $p,q\in P$, let $p\vee q\in P\cup\{\infty\}$ be their least upper bound or $\infty$ if $p,q$ have no upper bound in $P$. Let $(A,\mathcal{E}_{p},\mu_{p,q})_{p,q\in P}$ be a product system over $P$. Let $(\omega_{p})_{p\in P}$ be a Toeplitz representation of the product system in a $\mathrm{C}^{*}$-algebra $B$. The representation is called Nica covariant if, for all $p,q\in P$,

[TABLE]

Here we use $\Theta_{p}(\mathbb{K}(\mathcal{E}_{p}))=\omega_{p}(\mathcal{E}_{p})\omega_{p}(\mathcal{E}_{p})^{*}$ by (2.5).

Lemma 3.11**.**

A Toeplitz representation $(\omega_{p})_{p\in P}$ of a product system over $(G,P)$ satisfies (3.10) if and only if

[TABLE]

Proof.

Equation (3.12) multiplied on the left by $\omega_{p}(\mathcal{E}_{p})$ and on the right by $\omega_{q}(\mathcal{E}_{q})^{*}$ becomes (3.10) because $\omega_{p}(\mathcal{E}_{p})\cdot\omega_{p^{-1}(p\vee q)}(\mathcal{E}_{p^{-1}(p\vee q)})=\omega_{p\vee q}(\mathcal{E}_{p\vee q})$ as in the proof of Proposition 3.3. Conversely, multiply (3.10) on the left by $\omega_{p}(\mathcal{E}_{p})^{*}$ and on the right by $\omega_{q}(\mathcal{E}_{q})$. We may simplify

[TABLE]

and similarly for $q$. And

[TABLE]

In this way, (3.10) implies (3.12). ∎

Now let $(G,P)$ be $(\mathbb{Z},\mathbb{N})$. If $p,q\in\mathbb{N}$, then $p^{-1}(p\vee q)=\max\{p,q\}-p$ is [math] if $p\geq q$ and $q-p$ if $p\leq q$. So (3.12) becomes $\omega_{p}(\mathcal{E}_{p})^{*}\cdot\omega_{q}(\mathcal{E}_{q})\subseteq\omega_{q-p}(\mathcal{E}_{q-p})$ if $p\leq q$ and $\omega_{p}(\mathcal{E}_{p})^{*}\cdot\omega_{q}(\mathcal{E}_{q})\subseteq\omega_{p-q}(\mathcal{E}_{p-q})^{*}$ if $p\geq q$. Proposition 3.1 shows that each of these conditions is equivalent to the conditions in Definition 1.1. Proposition 3.3 shows that (3.12) holds automatically for weak representations of a global product system. In other words, any Toeplitz representation of a product system over $\mathbb{N}$ is Nica covariant, which is [Fowler:Product_systems]*Remark 5.2.

4. The Toeplitz algebra

In this section, we define the Toeplitz $\mathrm{C}^{*}$-algebra as the universal $\mathrm{C}^{*}$-algebra for representations of a partial product system. We equip the Toeplitz $\mathrm{C}^{*}$-algebra with a gauge action of the circle group $\mathbb{T}$, describe the spectral subspaces, and prove a gauge-equivariant uniqueness theorem. We describe the fixed-point subalgebra of the gauge action as an inductive limit and then show that the Fock representation generates a faithful representation of the Toeplitz $\mathrm{C}^{*}$-algebra.

Definition 4.1**.**

Let $\mathcal{E}=(A,(\mathcal{E}_{n})_{n},\mu_{n,m})$ be a partial product system. Its Toeplitz algebra is a $\mathrm{C}^{*}$-algebra $\mathcal{T}$ with a representation $(\bar{\omega}_{n})_{n\in\mathbb{N}}$ that is universal in the following sense: for any representation $(\omega_{n})_{n\in\mathbb{N}}$ of $\mathcal{E}$ in a $\mathrm{C}^{*}$-algebra $B$, there is a unique ∗-homomorphism $\varrho\colon\mathcal{T}\to B$ with $\varrho\circ\bar{\omega}_{n}=\omega_{n}$ for all $n\in\mathbb{N}$.

For a product system in the usual sense, any weak representation is a representation by Proposition 3.3. Hence the universal property of the Toeplitz $\mathrm{C}^{*}$-algebra is the usual one in this case. So our definition of the Toeplitz $\mathrm{C}^{*}$-algebra generalises the usual definition for product systems.

Proposition 4.2**.**

Any partial product system $(A,\mathcal{E}_{n},\mu_{n,m})$ has a Toeplitz algebra. It is the universal $\mathrm{C}^{*}$-algebra generated by the symbols $\bar{\omega}_{n}(x)$ for $n\in\mathbb{N}$, $x\in\mathcal{E}_{n}$, subject to the following relations:

1. (1)

$\bar{\omega}_{0}$* is a ∗-homomorphism;* 2. (2)

the maps $x\mapsto\bar{\omega}_{n}(x)$ are linear for all $n\in\mathbb{N}_{\geq 1}$; 3. (3)

$\bar{\omega}_{n}(x)\bar{\omega}_{m}(y)=\bar{\omega}_{n+m}(\mu_{n,m}(x\otimes y))$* for $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$, $n,m\in\mathbb{N}$;* 4. (4)

$\bar{\omega}_{n}(x)^{*}\bar{\omega}_{n}(y)=\bar{\omega}_{0}(\langle x\,|\,\mathopen{}y\rangle_{A})$* for all $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{n}$ $n\in\mathbb{N}$;* 5. (5)

$\omega_{n}(x)^{*}\omega_{m}(y)=\omega_{m-n}(S_{n}(x)^{*}(y))$* for all $n,m\in\mathbb{N}$ with $m>n$ and $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$.*

Proof.

Let $\mathfrak{T}$ be the ∗-algebra with the generators and relations as above. Any $\mathrm{C}^{*}$-norm on $\mathfrak{T}$ satisfies

[TABLE]

where the inequality uses that $\bar{\omega}_{0}$ is a ∗-homomorphism. Hence the set of all $\mathrm{C}^{*}$-norms on $\mathfrak{T}$ has a maximum. We claim that the completion of $\mathfrak{T}$ in this maximal $\mathrm{C}^{*}$-norm is the Toeplitz algebra of the partial product system. The relations that define $\mathfrak{T}$ are exactly those that are needed to make $\bar{\omega}_{n}$ a representation of our partial product system in $\mathcal{T}$. Here we use Proposition 3.1, which shows that (5) implies $\omega_{n}(x)^{*}\omega_{m}(y)=\omega_{n-m}(S_{m}(y)^{*}(x))^{*}$ for $m<n$, $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$. Any representation $(\omega_{n})_{n\in\mathbb{N}}$ of the partial product system in a $\mathrm{C}^{*}$-algebra $B$ induces a unique ∗-homomorphism $\mathcal{T}\to B$ mapping $\bar{\omega}_{n}(x)\mapsto\omega_{n}(x)$ for all $n\in\mathbb{N}$, $x\in\mathcal{E}_{n}$. ∎

Next we define a gauge action on $\mathcal{T}$. The maps

[TABLE]

form a representation of the partial product system because $\bar{\omega}_{n}$ is one. By the universal property of $\mathcal{T}$, there is a unique ∗-homomorphism $\alpha\colon\mathcal{T}\to\mathrm{C}(\mathbb{T},\mathcal{T})$ with $\alpha(\bar{\omega}_{n}(x))=\alpha_{n}(x)$ for all $n\in\mathbb{N}$, $x\in\mathcal{E}_{n}$. Define $\alpha_{t}\colon\mathcal{T}\to\mathcal{T}$ for $t\in\mathbb{T}$ by $\alpha_{t}\mathrel{\vcentcolon=}\mathrm{ev}_{t}\circ\alpha$. Since $\alpha$ is a ∗-homomorphism, each $\alpha_{t}$ is a ∗-homomorphism and $t\mapsto\alpha_{t}(x)$ is continuous for each $x\in\mathcal{T}$. Since $\alpha_{t}(\alpha_{s}(\bar{\omega}_{n}(x)))=(ts)^{n}\bar{\omega}_{n}(x)=\alpha_{ts}(\bar{\omega}_{n}(x))$ and $\alpha_{1}(\bar{\omega}_{n}(x))=\bar{\omega}_{n}(x)=\mathrm{id}_{\mathcal{T}}(\bar{\omega}_{n}(x))$ for $x\in\mathcal{E}_{n}$ and $n\in\mathbb{N}$, the uniqueness part of the universal property of $\mathcal{T}$ implies $\alpha_{t}\alpha_{s}=\alpha_{ts}$ for all $t,s\in\mathbb{T}$ and $\alpha_{1}=\mathrm{id}_{\mathcal{T}}$. Thus each $\alpha_{t}$ is bijective and $\mathbb{T}\ni t\mapsto\alpha_{t}$ is a continuous action of $\mathbb{T}$ on $\mathcal{T}$ by ∗-automorphisms.

A circle action on a $\mathrm{C}^{*}$-algebra gives a lot of useful extra structure (see [Exel:Circle_actions]). We now use this for the gauge action on $\mathcal{T}$. We define its spectral subspaces

[TABLE]

for $n\in\mathbb{Z}$. These spectral subspaces form a so-called $\mathbb{Z}$-grading, that is, $\mathcal{T}_{n}\mathcal{T}_{m}\subseteq\mathcal{T}_{n+m}$ and $\mathcal{T}_{n}^{*}=\mathcal{T}_{-n}$ for all $n,m\in\mathbb{Z}$, and the closed linear span of the subspaces $\mathcal{T}_{n}$ is dense in $\mathcal{T}$. In particular,

[TABLE]

is a $\mathrm{C}^{*}$-subalgebra, the fixed-point subalgebra of $\alpha$.

We define the spectral projections $E_{n}\colon\mathcal{T}\to\mathcal{T}$ for $n\in\mathbb{Z}$ by

[TABLE]

for $x\in\mathcal{T}$, where $\mathrm{d}t$ denotes the normalised Haar measure on the compact group $\mathbb{T}$. Each $E_{n}$ is norm contractive and is an idempotent operator with image $\mathcal{T}_{n}$ that vanishes on $\mathcal{T}_{m}$ for $m\neq n$. In particular, $E_{0}\colon\mathcal{T}\to\mathcal{T}_{0}$ is a conditional expectation. It is well known to be faithful, that is, if $x\in\mathcal{T}$ satisfies $x\geq 0$ and $E_{0}(x)=0$, then $x=0$. This implies the following gauge-invariant uniqueness theorem:

Theorem 4.3**.**

Let $B$ be a $\mathrm{C}^{*}$-algebra with an action $\beta$ of $\mathbb{T}$. Let $\varrho\colon\mathcal{T}\to B$ be a $\mathbb{T}$-equivariant ∗-homomorphism. If $\varrho$ is injective on $\mathcal{T}_{0}\subseteq\mathcal{T}$, then it is injective on $\mathcal{T}$.

Proof.

Let $\varrho(x)=0$ for some $x\in\mathcal{T}$. Let $E^{B}_{0}$ denote the [math]th spectral projection on $B$. Since $\varrho$ is $\mathbb{T}$-equivariant, it intertwines the spectral projections $E_{0}$ on $\mathcal{T}$ and $E_{0}^{B}$ on $B$. So

[TABLE]

This implies $E_{0}(x^{*}x)=0$ because $E_{0}(x^{*}x)\in\mathcal{T}_{0}$, and $\varrho|_{\mathcal{T}_{0}}$ is injective. Then $x=0$ because $E_{0}$ is faithful. ∎

Let $\bar{\Theta}_{m,n}\colon\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})\to\mathcal{T}$ be the maps defined by the representation $\bar{\omega}_{n}$ as in (2.5), that is, $\bar{\Theta}_{m,n}(\lvert x\rangle\langle y\rvert)=\bar{\omega}_{n}(x)\bar{\omega}_{m}(y)^{*}$ for $n,m\in\mathbb{N}$, $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$. The relations in Proposition 4.2 imply that any word in the generators of $\mathfrak{T}$ may be reduced to one of the form $\bar{\omega}_{n}(x)\bar{\omega}_{m}(y)^{*}$ for some $n,m\in\mathbb{N}$, $x\in\mathcal{E}_{n}$, $y\in\mathcal{E}_{m}$. Therefore, the closed linear span of $\bar{\Theta}_{m,n}(\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n}))$ for $n,m\in\mathbb{N}$ is a dense subspace in $\mathcal{T}$. It is a ∗-subalgebra by Proposition 3.1.(7).

Lemma 4.4**.**

The spectral subspace $\mathcal{T}_{k}\subseteq\mathcal{T}$ for $k\in\mathbb{Z}$ is the closed linear span of $\bar{\Theta}_{n,n+k}(\mathbb{K}(\mathcal{E}_{n},\mathcal{E}_{n+k}))$ for all $n\in\mathbb{N}$ with $n\geq-k$.

Proof.

If $x\in\mathcal{E}_{n+k}$, $y\in\mathcal{E}_{n}$, then $\alpha_{t}(\bar{\omega}_{n+k}(x)\bar{\omega}_{n}(y)^{*})=t^{k}\bar{\omega}_{n+k}(x)\bar{\omega}_{n}(y)^{*}$. Thus $\bar{\omega}_{n+k}(x)\bar{\omega}_{n}(y)^{*}\in\mathcal{T}_{k}$. Now (2.5) implies that $\bar{\Theta}_{n,n+k}(\mathbb{K}(\mathcal{E}_{n},\mathcal{E}_{n+k}))$ is contained in $\mathcal{T}_{k}$. We have already observed that the linear span of $\bar{\Theta}_{n,m}(\mathbb{K}(\mathcal{E}_{n},\mathcal{E}_{m}))$ is dense in $\mathcal{T}$. If $x\in\mathcal{T}_{k}$ and $\varepsilon>0$, then there is a finite linear combination $\sum_{n,m\in\mathbb{N}}\bar{\Theta}_{n,m}(x_{n,m})$ with $x_{n,m}\in\mathbb{K}(\mathcal{E}_{n},\mathcal{E}_{m})$ that is $\varepsilon$-close to $x$. Then

[TABLE]

is still $\varepsilon$-close to $x$. So the closed linear span of $\bar{\Theta}_{n,n+k}(\mathbb{K}(\mathcal{E}_{n},\mathcal{E}_{n+k}))$ is dense in $\mathcal{T}_{k}$. ∎

Theorem 4.5**.**

The map

[TABLE]

is injective and its image $\mathcal{T}_{0,N}$ is a $\mathrm{C}^{*}$-subalgebra of $\mathcal{T}_{0}$. These $\mathrm{C}^{*}$-subalgebras for $N\to\infty$ form an inductive system with colimit $\mathcal{T}_{0}$.

Proof.

The Fock representation $(S_{n})_{n\in\mathbb{N}}$ induces a ∗-homomorphism $\varphi\colon\mathcal{T}\to\mathbb{B}(\mathcal{F})$ with $\varphi(\bar{\omega}_{n}(x))=S_{n}(x)$ for all $n\in\mathbb{N}$, $x\in\mathcal{E}_{n}$. Define the ∗-homomorphisms $\Theta_{n}\colon\mathbb{K}(\mathcal{E}_{n})\to\mathbb{B}(\mathcal{F})$ as in (2.5). Then $\varphi(\bar{\Theta}_{n}(x))=\Theta_{n}(x)$ for all $n\in\mathbb{N}$, $x\in\mathbb{K}(\mathcal{E}_{n})$. The operator $\Theta_{n}(x)$ is block diagonal on $\mathcal{F}\mathrel{\vcentcolon=}\bigoplus_{j=0}^{\infty}\mathcal{E}_{j}$. Its first $n-1$ diagonal entries are [math], and the $n$th diagonal entry is $x$ acting on $\mathcal{E}_{n}$. Let $x_{j}\in\mathbb{K}(\mathcal{E}_{j})$ for $j=0,\dotsc,N$ satisfy $\sum_{j=0}^{N}\Theta_{j}(x_{j})=0$. We prove recursively that $x_{0}=0$, $x_{1}=0$, $x_{2}=0$, and so on, by looking at the $j$th diagonal entry for $j=0,\dotsc,N$. So the composite map

[TABLE]

is injective. Even more, the recursive proof above shows that this injective map has a bounded inverse on its image. So its image is closed in $\mathcal{T}_{0}$. Equation (3.2) and Lemma 4.4 imply that $\mathcal{T}_{0,N}$ is a ∗-subalgebra in $\mathcal{T}_{0}$. Since it is closed as well, it is a $\mathrm{C}^{*}$-subalgebra. Lemma 4.4 shows that the union $\bigcup_{N=0}^{\infty}\mathcal{T}_{0,N}$ is dense in $\mathcal{T}_{0}$. Hence $\mathcal{T}_{0}$ is the inductive limit $\mathrm{C}^{*}$-algebra of the inductive system $(\mathcal{T}_{0,N})_{N\in\mathbb{N}}$. ∎

Theorem 4.6**.**

The Fock representation $(S_{n})_{n\in\mathbb{N}}$ on the Fock module $\mathcal{F}$ over $A$ induces a faithful representation of $\mathcal{T}$. So $\mathcal{T}$ is isomorphic to the $\mathrm{C}^{*}$-subalgebra of $\mathbb{B}(\mathcal{F})$ generated by $S_{n}(\mathcal{E}_{n})$ for all $n\in\mathbb{N}$.

Proof.

Let $\varphi\colon\mathcal{T}\to\mathbb{B}(\mathcal{F})$ be the representation induced by the Fock representation; this exists because $\mathcal{E}$ is a partial product system. The proof of Theorem 4.5 shows that the restriction of $\varphi$ to $\mathcal{T}_{0,N}\subseteq\mathcal{T}$ is injective for all $N\in\mathbb{N}$. Since $\mathcal{T}_{0}$ is the inductive limit of these $\mathrm{C}^{*}$-subalgebras by Theorem 4.5, it follows that $\varphi|_{\mathcal{T}_{0}}$ is injective. The Fock Hilbert module carries an obvious gauge action with spectral subspaces $\mathcal{F}_{n}=\mathcal{E}_{n}$. Let $\beta\colon\mathbb{T}\to\mathbb{B}(\mathcal{F})$ be the induced action. The Fock representation $\varphi$ is $\mathbb{T}$-equivariant because $S_{n}(x)$ belongs to the $n$th spectral subspace of $\mathbb{B}(\mathcal{F})$ for all $n\in\mathbb{N}$, $x\in\mathcal{E}_{n}$. Hence Theorem 4.3 shows that $\varphi$ is injective. Its image is the $\mathrm{C}^{*}$-subalgebra generated by the operators $S_{n}(x)$ for $n\in\mathbb{N}$, $x\in\mathcal{E}_{n}$ because the elements $\bar{\omega}_{n}(x)$ for $n\in\mathbb{N}$, $x\in\mathcal{E}_{n}$ generate $\mathcal{T}$. ∎

Proposition 4.7**.**

The map $\bigoplus_{m,n\in\mathbb{N}}\bar{\Theta}_{m,n}\colon\bigoplus_{m,n\in\mathbb{N}}\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})\to\mathcal{T}$ is injective.

Proof.

It suffices to prove that $\bigoplus_{m,n}\Theta_{m,n}\colon\bigoplus_{m,n}\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})\to\mathcal{T}\hookrightarrow\mathbb{B}(\mathcal{F})$ is injective. We describe operators on $\mathcal{F}$ by block matrices. If $x\in\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})$, then the $j,k$-entry of $\Theta_{m,n}(x)$ vanishes unless $j-k=n-m$ and $k\geq m$, and the $n,m$-entry is $x\colon\mathcal{E}_{m}\to\mathcal{E}_{n}$. Let $\sum_{m,n\in\mathbb{N}}\Theta_{m,n}(x_{m,n})=0$ for some $x_{m,n}\in\mathbb{K}(\mathcal{E}_{m},\mathcal{E}_{n})$ with only finitely many non-zero $x_{n,m}$. Now we examine the $j,k$-entries of $\sum_{m,n\in\mathbb{N}}\Theta_{m,n}(x_{m,n})$ for increasing $j$. For $j=0$, we see that $x_{m,0}=0$ for all $m\in\mathbb{N}$. An induction over $j$ shows that $x_{m,j}=0$ for all $m\in\mathbb{N}$ and all $j\in\mathbb{N}$. ∎

4.1. Hereditary restrictions and quotients

Throughout this subsection, let $A$ be a $\mathrm{C}^{*}$-algebra and let $\mathcal{E}\mathrel{\vcentcolon=}(A,\mathcal{E}_{n},\mu_{n,m})_{n,m\in\mathbb{N}}$ be a partial product system over $A$. Let $\mathcal{T}$ be its Toeplitz $\mathrm{C}^{*}$-algebra. We are going to restrict the partial product system to a hereditary $\mathrm{C}^{*}$-subalgebra $H\subseteq A$ and a quotient $A/I$ for an invariant ideal $I$ in $A$. We show that the Toeplitz $\mathrm{C}^{*}$-algebra for the restriction to $A/I$ is a quotient of $\mathcal{T}$.

First let $H\subseteq A$ be a hereditary $\mathrm{C}^{*}$-subalgebra. Define

[TABLE]

the set of all products $a\cdot x\cdot b$ with $a,b\in H$, $x\in\mathcal{E}_{n}$ is already a closed linear subspace of $\mathcal{E}_{n}$ by the Cohen–Hewitt Factorisation Theorem. The multiplication maps in the given partial product system $(\mathcal{E}_{n})_{n\in\mathbb{N}}$ restrict to multiplication maps

[TABLE]

Lemma 4.8**.**

The Hilbert $H,H$-modules $\mathcal{E}_{n}|_{H}$ and the multiplication maps $\mu_{n,m}|_{H}$ form a partial product system $\mathcal{E}|_{H}$ over $H$.

Proof.

Let $\mathcal{F}$ be $\bigoplus_{n=0}^{\infty}\mathcal{E}_{n}$ with the Fock representation $(S_{n})_{n\in\mathbb{N}}$ of $\mathcal{E}$. The Hilbert submodule $\mathcal{F}\cdot H=\bigoplus_{n=0}^{\infty}\mathcal{E}_{n}\cdot H$ is invariant for $S_{n}(x)$ and $S_{n}(x)^{*}$ for all $x\in\mathcal{E}_{n}$. The Hilbert submodule $H\cdot\mathcal{F}\cdot H=\bigoplus_{n=0}^{\infty}H\cdot\mathcal{E}_{n}\cdot H$ is still invariant for $S_{n}(x)$ and $S_{n}(x)^{*}$ for all $x\in H\cdot\mathcal{E}_{n}\cdot H$. We claim that $S^{\prime}_{n}(x)=S_{n}(x)|_{H\cdot\mathcal{F}\cdot H}$ for $n\in\mathbb{N}$