A uniqueness theorem for the Nica-Toeplitz algebra of a compactly aligned product system

TL;DR

This paper proves a uniqueness theorem for Nica-Toeplitz algebras associated with compactly aligned product systems, clarifying conditions under which these algebras are uniquely determined by their representations.

Contribution

It establishes a new uniqueness theorem for Nica-Toeplitz algebras of compactly aligned product systems, extending previous results in the theory of $C^*$-algebras.

Findings

Proves a uniqueness theorem for Nica-Toeplitz algebras

Clarifies conditions for algebraic uniqueness

Extends understanding of product system $C^*$-algebras

Abstract

Fowler introduced the notion of a product system: a collection of Hilbert bimodules $\mathbf{X}=\left\{\mathbf{X}_p:p\in P\right\}$ indexed by a semigroup $P$, endowed with a multiplication implementing isomorphisms $\mathbf{X}_p\otimes_A \mathbf{X}_q\cong \mathbf{X}_{pq}$. When $P$ is quasi-lattice ordered, Fowler showed how to associate a $C^*$-algebra $\mathcal{NT}_\mathbf{X}$ to $\mathbf{X}$, generated by a universal representation satisfying some covariance condition. In this article we prove a uniqueness theorem for these so called Nica-Toeplitz algebras.

Equations

$${\iota }_p^{pq}(S):=M_{p,q}\circ \left(S{\otimes }_A{\mathrm{id}}_{{\mathbf{X}}_q}\right)\circ M_{p,q}^{-1}$$

$${\psi }^{(p)}(S){\psi }^{(q)}(T)=\{\begin{array}{ll}{\psi }^{(p\vee q)}\left({\iota }_p^{p\vee q}(S){\iota }_q^{p\vee q}(T)\right)& \text{if p∨q<∞}\\ 0& \text{otherwise.}\end{array}$$

$${\psi }_p({\mathbf{X}}_p{)}^{\ast }{\psi }_q({\mathbf{X}}_q)\in \{\begin{array}{ll}\overline{\mathrm{span}}\{{\psi }_{p^{-1}(p\vee q)}({\mathbf{X}}_{p^{-1}(p\vee q)}){\psi }_{q^{-1}(p\vee q)}({\mathbf{X}}_{q^{-1}(p\vee q)}{)}^{\ast }\}& \text{if p∨q<∞}\\ \{0\}& \text{if p∨q=∞.}\end{array}$$

$$\displaystyle A\ni a\mapsto\psi_{e}(a)\prod_{t\in K}\big{(}\mathrm{id}_{\mathcal{H}}-P_{t}^{\psi}\big{)}\in\mathcal{B}(\mathcal{H})$$

$${\rho }_p^{\psi }(S)\left({\psi }_p(x)h\right)={\psi }_p(Sx)h\text{for each x∈Xp, h∈H}$$

$$⟨{\psi }_p(x)h,{\psi }_p(y)k{⟩}_{\mathbb{C}}=⟨h,{\psi }_p(x{)}^{\ast }{\psi }_p(y)k{⟩}_{\mathbb{C}}=⟨h,{\psi }_e\left(⟨x,y{⟩}_A\right)k{⟩}_{\mathbb{C}}$$

$$⟨U(y{\otimes }_{A}k),f{⟩}_{\mathbb{C}}=⟨{\psi }_p(y)k,f{⟩}_{\mathbb{C}}=0,$$

$${\rho }_p^{\psi }(S):=U\circ \left({\mathbf{X}}_p\text{-}{\mathrm{Ind}}_A^{{\mathcal{L}}_A\left({\mathbf{X}}_p\right)}{\psi }_e(S)\right)\circ U^{\ast }$$

$${\rho }_p^{\psi }(S)({\psi }_p(x)h)$$

$$=\left(U\circ \left({\mathbf{X}}_p\text{-}{\mathrm{Ind}}_A^{{\mathcal{L}}_A\left({\mathbf{X}}_p\right)}{\psi }_e(S)\right)\right)(x{\otimes }_{A}h)=U\left(Sx{\otimes }_{A}h\right)={\psi }_p(Sx)h.$$

$${\rho }_p^{\psi }\left({\Theta }_{x,y}\right)\left({\psi }_p(z)h\right)$$

$$={\psi }_p(x){\psi }_p(y{)}^{\ast }{\psi }_p(z)h={\psi }^{(p)}\left({\Theta }_{x,y}\right)\left({\psi }_p(z)h\right).$$

$$\big{\langle}\psi^{(p)}\left(\Theta_{x,y}\right)h,k\big{\rangle}_{\mathbb{C}}=\langle\psi_{p}(x)\psi_{p}(y)^{*}h,k\rangle_{\mathbb{C}}=\langle h,\psi_{p}(y)\psi_{p}(x)^{*}k\rangle_{\mathbb{C}}=0$$

$${\rho }_e^{\psi }(a)P_q={\rho }_e^{\psi }(a)={\rho }_e^{\psi }\left({\iota }_e^e(a)\right)={\rho }_e^{\psi }\left({\iota }_e^q(a)\right).$$

$${\rho }_q^{\psi }\left({\iota }_e^q(a)\right)\left({\psi }_q(x)h\right)$$

$$={\rho }_e^{\psi }(a)\left({\psi }_q(x)h\right)={\rho }_e^{\psi }(a)P_q^{\psi }\left({\psi }_q(x)h\right)$$

$${\rho }_{pq}^{\psi }\left({\iota }_p^{pq}(S)\right)\left({\psi }_{pq}(xy)h\right)={\psi }_{pq}\left({\iota }_p^{pq}(S)(xy)\right)h={\psi }_{pq}\left((Sx)y\right)h={\psi }_p(Sx){\psi }_q(y)h,$$

$${\rho }_p^{\psi }(S)\left({\psi }_p(x){\psi }_q(y)h\right)={\rho }_p^{\psi }(S)\left({\psi }_{pq}(xy)h\right)={\rho }_p^{\psi }(S)P_{pq}^{\psi }\left({\psi }_{pq}(xy)h\right).$$

$$P_p^{\psi }{P}_q^{\psi }=P_{p\vee q}^{\psi }.$$

$${\Vert e_j{\Theta }_{x^{\prime },x^{\prime }}-{\Theta }_{x^{\prime },x^{\prime }}\Vert }_{{\mathcal{K}}_A({\mathbf{X}}_p)}<\frac{\epsilon }{{\Vert x^{\prime }\Vert }_{{\mathbf{X}}_p}+1}.$$

$${\Vert e_{j}x-x\Vert }_{{\mathbf{X}}_p}$$

$$={\Vert \left(e_j{\Theta }_{x^{\prime },x^{\prime }}-{\Theta }_{x^{\prime },x^{\prime }}\right)x^{\prime }\Vert }_{{\mathbf{X}}_p}\le {\Vert e_j{\Theta }_{x^{\prime },x^{\prime }}-{\Theta }_{x^{\prime },x^{\prime }}\Vert }_{{\mathcal{K}}_A({\mathbf{X}}_p)}{\Vert x^{\prime }\Vert }_{{\mathbf{X}}_p}<\epsilon .$$

$${\rho }_p^{\psi }(e_i)h=0={\rho }_p^{\psi }\left({\mathrm{id}}_{{\mathbf{X}}_p}\right)h$$

$$\bigg{\|}h-\sum_{i=1}^{n}\psi_{p}(x_{i})h_{i}\bigg{\|}_{\mathcal{H}}<\varepsilon/4.$$

$$\displaystyle\bigg{\|}\rho_{p}^{\psi}\left(e_{i}-\mathrm{id}_{\mathbf{X}_{p}}\right)\bigg{(}h-\sum_{i=1}^{n}\psi_{p}(x_{i})h_{i}\bigg{)}\bigg{\|}_{\mathcal{H}}$$

$$\displaystyle\leq\left\|e_{i}-\mathrm{id}_{\mathbf{X}_{p}}\right\|_{\mathcal{L}_{A}(\mathbf{X}_{p})}\bigg{\|}h-\sum_{i=1}^{n}\psi_{p}(x_{i})h_{i}\bigg{\|}_{\mathcal{H}}$$

$$\displaystyle\leq 2\bigg{\|}h-\sum_{i=1}^{n}\psi_{p}(x_{i})h_{i}\bigg{\|}_{\mathcal{H}}$$

$$<\epsilon /2.$$

$${\Vert e_k{x}_i-x_i\Vert }_{{\mathbf{X}}_p}<\frac{\epsilon }{2n\left({\max }_{1\le i\le n}{\Vert h_i\Vert }_{\mathcal{H}}+1\right)}.$$

$$\displaystyle\bigg{\|}\rho_{p}^{\psi}\left(e_{k}-\mathrm{id}_{\mathbf{X}_{p}}\right)\bigg{(}\sum_{i=1}^{n}\psi_{p}(x_{i})h_{i}\bigg{)}\bigg{\|}_{\mathcal{H}}$$

$$\le \sum _{i=1}^n{\Vert {\psi }_p\left(\left(e_k-{\mathrm{id}}_{{\mathbf{X}}_p}\right)x\right)\Vert }_{\mathcal{B}(\mathcal{H})}{\Vert h_i\Vert }_{\mathcal{H}}$$

$$\le \sum _{i=1}^n{\Vert \left(e_k-{\mathrm{id}}_{{\mathbf{X}}_p}\right)x\Vert }_{{\mathbf{X}}_p}{\Vert h_i\Vert }_{\mathcal{H}}$$

$$<\sum _{i=1}^n\frac{\epsilon {\Vert h_i\Vert }_{\mathcal{H}}}{2n\left({\max }_{1\le i\le n}{\Vert h_i\Vert }_{\mathcal{H}}+1\right)}$$

$$\le \epsilon /2.$$

$$\displaystyle\big{\|}\rho_{p}^{\psi}$$

Full text

This research was supported by an Australian Government Research Training Program (RTP) Scholarship and by the Marsden grant 15-UOO-071 from the Royal Society of New Zealand.

1. Introduction

Suppose $A$ is a $C^{*}$-algebra and $X$ is a right Hilbert $A$-module. When $X$ comes equipped with a left action of $A$ by adjointable operators, we call $X$ a Hilbert $A$-bimodule. When this left action of $A$ is faithful, Pimsner showed how to associate a $C^{*}$-algebra $\mathcal{T}_{X}$ to $X$, called the Toeplitz algebra of $X$, generated by the raising and lowering operators on the Fock space of $X$ [14]. In [7], Fowler and Raeburn generalised the situation to arbitrary left actions by adjointable operators, and showed that $\mathcal{T}_{X}$ may be realised as the universal $C^{*}$-algebra generated by so called Toeplitz representations. In the spirit of Coburn’s Theorem for the classical Toeplitz algebra [2], Fowler and Raeburn also proved a uniqueness theorem for $\mathcal{T}_{X}$ [7, Theorem 2.1] that provides a sufficient condition for a representation of $\mathcal{T}_{X}$ to be faithful. Loosely speaking, their result states that a representation of $\mathcal{T}_{X}$ on a Hilbert space $\mathcal{H}$ will be faithful, provided the representation leaves enough room in $\mathcal{H}$ for the coefficient algebra $A$ to act faithfully.

Subsequently Fowler introduced the notion of a product system of Hilbert bimodules [6], generalising the continuous product systems of Hilbert spaces studied by Arveson [1] and the discrete product systems studied by Dinh [3]. Loosely speaking, a product system of Hilbert $A$-bimodules over a unital semigroup $P$ consists of a semigroup $\mathbf{X}=\bigsqcup_{p\in P}\mathbf{X}_{p}$, such that each $\mathbf{X}_{p}$ is a Hilbert $A$-bimodule, and the map $x\otimes_{A}y\mapsto xy$ extends to an isomorphism from $\mathbf{X}_{p}\otimes_{A}\mathbf{X}_{q}$ to $\mathbf{X}_{pq}$ for each $p,q\in P\setminus\{e\}$. Motivated by the work of Nica [13] and Laca and Raeburn on Toeplitz algebras associated to non-abelian groups [10], Fowler studied representations of compactly aligned product systems over quasi-lattice ordered groups — semigroups sitting inside groups possessing a semi-lattice like structure, satisfying an additional constraint called Nica covariance. Generalising the Toeplitz algebra associated to a single Hilbert bimodule, Fowler showed how to associate a $C^{*}$-algebra $\mathcal{NT}_{\mathbf{X}}$, generated by a universal Nica covariant representation, to each compactly aligned product system $\mathbf{X}$. We call this $C^{*}$-algebra the Nica–Toeplitz algebra of $\mathbf{X}$. Furthermore, Fowler associates a twisted semigroup crossed product algebra to each compactly aligned product system, and characterises their faithful representations [6, Theorem 7.2]. Restricting to the subalgebra $\mathcal{NT}_{\mathbf{X}}$ then gives a uniqueness theorem for Nica–Toeplitz algebras, generalising both Laca and Raeburn’s uniqueness theorem for Toeplitz algebras of quasi-lattice ordered groups [10, Theorem 3.7] and Fowler and Raeburn’s uniqueness theorem for Toeplitz algebras of Hilbert bimodules [7, Theorem 2.1] — at least in the case where the bimodule is ‘essential’.

In this article we prove a slightly more general version of Fowler’s uniqueness theorem for Nica–Toeplitz algebras associated to compactly aligned product systems over quasi-lattice ordered groups. The result gives a sufficient condition for the induced representation $\psi_{*}$ of a Nica covariant representation $\psi$ (on a Hilbert space $\mathcal{H}$) to be faithful. This condition basically says that the ranges of all the operators $\left\{\psi_{p}(x):x\in\mathbf{X}_{p},\ p\in P\setminus\{e\}\right\}$ should leave enough room in $\mathcal{H}$ for $A$ to act faithfully. When $A$ acts by compacts on each fibre of $\mathbf{X}$, we will see that this condition is also necessary. Unlike in Fowler’s result, we do not insist that each fibre of the product system is essential (i.e. we do not require that $\mathbf{X}_{p}=\overline{A\cdot\mathbf{X}_{p}}$ for each $p\in P$). Whilst there do not seem to many ‘natural’ examples of Hilbert bimodules with nondegenerate left actions, this level of generality was made use of in our article [5]. Moreover, in contrast to Fowler’s proof, we do not view the Nica–Toeplitz algebra as a subalgebra of a twisted semigroup crossed product, working with $\mathcal{NT}_{\mathbf{X}}$ directly. Furthermore, we note that Fowler’s result as stated in [6, Theorem 7.2] is technically false when applied to any product system over the trivial semigroup $\{e\}$, and correct this error in our result.

The article is set out as follows. In Section 2 we fix notation and recap the necessary background material for Hilbert bimodules, product systems of Hilbert bimodules, and their associated Nica–Toeplitz algebras. In Section 3 we present the proof of our uniqueness theorem for Nica-Toeplitz algebras.

After completing this article, it was brought to our attention that Kwaśniewski and Larsen had already proved a far-reaching generalisation of our (along with Fowler’s) uniqueness theorem [8, Corollary 10.14] for full Nica–Toeplitz algebras associated to well-aligned ideals of right tensor $C^{*}$-precategories. Subsequent to the initial version of this article appearing on the arXiv, Kwaśniewski and Larsen showed how their more general result can be applied to product systems over right LCM semigroups (themselves a generalisation of quasi-lattice ordered groups) [9, Theorem 2.19]. Despite this, we still feel that the results in this article will find use amongst those studying product systems over quasi-lattice ordered groups. We provide a direct proof of the uniqueness theorem in the quasi-lattice ordered case, and as such avoid the various technical conditions present in [9, Theorem 2.19]. We also note that in Corollary 3.12, we show how to extend the uniqueneness theorem to representations in arbitrary $C^{*}$-algebras (rather than just on Hilbert spaces), provided the action on each fibre is compact.

2. Preliminaries

2.1. Hilbert bimodules

We attempt to summarise only those aspects of Hilbert bimodules that we will need. Readers unfamiliar with this material, or looking for more detail, are directed to [11].

Let $A$ be a $C^{*}$-algebra. A (right) inner-product $A$-module is a complex vector space $X$ equipped with a map $\langle\cdot,\cdot\rangle_{A}:X\times X\rightarrow A$, linear in its second argument, and a right action of $A$, such that for any $x,y\in X$ and $a\in A$, we have

1. (i)

$\langle x,y\rangle_{A}=\langle y,x\rangle_{A}^{*}$; 2. (ii)

$\langle x,y\cdot a\rangle_{A}=\langle x,y\rangle_{A}a$; 3. (iii)

$\langle x,x\rangle_{A}\geq 0$ in $A$; and 4. (iv)

$\langle x,x\rangle_{A}=0$ if and only if $x=0$.

It follows from [11, Proposition 1.1] that the formula $\left\|x\right\|_{X}:=\left\|\langle x,x\rangle_{A}\right\|_{A}^{1/2}$ defines a norm on $X$. If $X$ is complete with respect to this norm, we say that $X$ is a (right) Hilbert $A$-module.

Let $X$ be a (right) Hilbert $A$-module. We say that a map $T:X\rightarrow X$ is adjointable if there exists a map $T^{*}:X\rightarrow X$ such that $\langle Tx,y\rangle_{A}=\langle x,T^{*}y\rangle_{A}$ for each $x,y\in X$. Every adjointable operator $T$ is automatically linear and continuous, and the adjoint $T^{*}$ is unique. The collection of adjointable operators on $X$, denoted by $\mathcal{L}_{A}(X)$, equipped with the operator norm is a $C^{*}$-algebra. For each $x,y\in X$ there is an adjointable operator $\Theta_{x,y}\in\mathcal{L}_{A}(X)$ defined by $\Theta_{x,y}(z)=x\cdot\langle y,z\rangle_{A}$. We call operators of this form (generalised) rank-one operators. The closed subspace $\mathcal{K}_{A}(X):=\overline{\mathrm{span}}\{\Theta_{x,y}:x,y\in X\}$ is an essential ideal of $\mathcal{L}_{A}(X)$, whose elements we refer to as (generalised) compact operators.

A (right) Hilbert $B$–$A$-bimodule is a (right) Hilbert $A$-module $X$ together with a $*$-homomorphism $\phi:B\rightarrow\mathcal{L}_{A}(X)$. When $A=B$, we say that $X$ is a Hilbert $A$-bimodule. We think of $\phi$ as implementing a left action of $B$ on $X$, and frequently write $b\cdot x$ for $\phi(b)(x)$. Since each $\phi(b)\in\mathcal{L}_{A}(X)$ is $A$-linear, we have that $b\cdot(x\cdot a)=(b\cdot x)\cdot a$ for each $a\in A$, $b\in B$, and $x\in X$.

An important example is the Hilbert $A$-bimodule ${}_{A}A_{A}$, which is just the set $A$ equipped with the inner product given by $\langle a,b\rangle_{A}=a^{*}b$ and left and right actions of $A$ given by multiplication. Then $\mathcal{K}_{A}({}_{A}A_{A})$ is isomorphic to $A$ via the map $\Theta_{a,b}\mapsto ab^{*}$, whilst $\mathcal{L}_{A}({}_{A}A_{A})$ is isomorphic to the multiplier algebra of $A$.

The Hewitt–Cohen–Blanchard factorisation theorem [17, Proposition 2.31] says that if $x$ is an element of a Hilbert $A$-module $X$, then there exists a unique $x^{\prime}\in X$ such that $x=x^{\prime}\cdot\langle x^{\prime},x^{\prime}\rangle_{A}$. Hence, $X$ is a right nondegenerate $A$-module in the sense that $X=\overline{\mathrm{span}}\{x\cdot a:x\in X,a\in A\}$. It is not necessarily true that every Hilbert $A$-bimodule is left nondegenerate in the sense that $X=\overline{\mathrm{span}}\{a\cdot x:x\in X,a\in A\}$ (in [6], Fowler calls such bimodules essential).

The balanced tensor product $X\otimes_{A}Y$ of two Hilbert $A$-bimodules $X$ and $Y$ is formed as follows. Let $X\odot Y$ be the algebraic tensor product of $X$ and $Y$ as complex vector spaces, and let $X\odot_{A}Y$ be the quotient of $X\odot Y$ by the subspace spanned by elements of the form $x\cdot a\odot y-x\odot a\cdot y$ where $x\in X$, $y\in Y$, and $a\in A$ (we write $x\odot_{A}y$ for the coset containing $x\odot y$). Then the formula $\langle x\odot_{A}y,w\odot_{A}z\rangle_{A}:=\langle y,\langle x,w\rangle_{A}\cdot z\rangle_{A},$ determines a bounded $A$-valued sesquilinear form on $X\odot_{A}Y$. Let $N$ be the subspace $\mathrm{span}\{n\in X\odot_{A}Y:\langle n,n\rangle_{A}=0\}$. The formula $\left\|z+N\right\|:=\inf_{n\in N}\left\|\langle z+n,z+n\rangle_{A}\right\|_{A}^{1/2}$ defines a norm on $(X\odot_{A}Y)/N$, and we define $X\otimes_{A}Y$ to be the completion of $(X\odot_{A}Y)/N$ with respect to this norm. The balanced tensor product $X\otimes_{A}Y$ carries a left and right action of $A$, such that $a\cdot(x\otimes_{A}y)\cdot b=(a\cdot x)\otimes_{A}(y\cdot b)$ for each $x\in X$, $y\in Y$, and $a,b\in A$.

Given Hilbert $A$-bimodules $X$ and $Y$ and an adjointable operator $S\in\mathcal{L}_{A}(X)$, there exists an adjointable operator $S\otimes_{A}\mathrm{id}_{Y}\in\mathcal{L}_{A}(X\otimes_{A}Y)$ (with adjoint $S^{*}\otimes_{A}\mathrm{id}_{Y}$) determined by the formula $(S\otimes_{A}\mathrm{id}_{Y})(x\otimes_{A}y)=(Sx)\otimes_{A}y$ for each $x\in X$ and $y\in Y$.

We will also make use of the theory of induced representations. Given a Hilbert $B$–$A$-bimodule $X$ and a nondegenerate representation $\pi:A\rightarrow\mathcal{B}(\mathcal{H})$ of $A$ on a Hilbert space $\mathcal{H}$, [17, Proposition 2.66] gives a representation $X\text{-}\mathrm{Ind}_{A}^{B}\pi:B\rightarrow\mathcal{B}\left(X\otimes_{A}\mathcal{H}\right)$ such that $\left(X\text{-}\mathrm{Ind}_{A}^{B}\pi\right)(b)(x\otimes_{A}h)=(b\cdot x)\otimes_{A}h$ for each $b\in B$, $x\in X$, and $h\in\mathcal{H}$.

2.2. Product systems of Hilbert bimodules and quasi-lattice ordered groups

Let $A$ be a $C^{*}$-algebra and $P$ a semigroup with identity $e$. A product system over $P$ with coefficient algebra $A$ is a semigroup $\mathbf{X}=\bigsqcup_{p\in P}\mathbf{X}_{p}$ such that

1. (i)

$\mathbf{X}_{p}\subseteq\mathbf{X}$ is a Hilbert $A$-bimodule for each $p\in P$; 2. (ii)

$\mathbf{X}_{e}$ is equal to the Hilbert $A$-bimodule ${}_{A}A_{A}$; 3. (iii)

For each $p,q\in P\setminus\{e\}$, there exists a Hilbert $A$-bimodule isomorphism $M_{p,q}:\mathbf{X}_{p}\otimes_{A}\mathbf{X}_{q}\rightarrow\mathbf{X}_{pq}$ satisfying $M_{p,q}(x\otimes_{A}y)=xy$ for each $x\in\mathbf{X}_{p}$ and $y\in\mathbf{X}_{q}$; 4. (iv)

Multiplication in $\mathbf{X}$ by elements of $\mathbf{X}_{e}=A$ implements the left and right actions of $A$ on each $\mathbf{X}_{p}$, i.e. $xa=x\cdot a$ and $ax=a\cdot x$ for each $a\in A$, $x\in\mathbf{X}_{p}$, and $p\in P$.

For each $p\in P$, we write $\phi_{p}:A\rightarrow\mathcal{L}_{A}(\mathbf{X}_{p})$ for the $*$-homomorphism that implements the left action of $A$ on $\mathbf{X}_{p}$, i.e. $\phi_{p}(a)(x)=a\cdot x=ax$ for each $a\in A$ and $x\in\mathbf{X}_{p}$. Since $\mathbf{X}$ is a semigroup, multiplication in $\mathbf{X}$ is associative. In particular, $\phi_{pq}(a)(xy)=(\phi_{p}(a)x)y$ for all $p,q\in P$, $a\in A$, $x\in\mathbf{X}_{p}$, and $y\in\mathbf{X}_{q}$. Also, for each $p\in P$, we write $\langle\cdot,\cdot\rangle_{A}^{p}$ for the $A$-valued inner-product on $\mathbf{X}_{p}$.

By (ii) and (iv), for each $p\in P$ there exist $A$-linear inner-product preserving maps $M_{p,e}:\mathbf{X}_{p}\otimes_{A}\mathbf{X}_{e}\rightarrow\mathbf{X}_{p}$ and $M_{e,p}:\mathbf{X}_{e}\otimes_{A}\mathbf{X}_{p}\rightarrow\mathbf{X}_{p}$ such that $M_{p,e}(x\otimes_{A}a)=xa=x\cdot a$ and $M_{e,p}(a\otimes_{A}x)=ax=a\cdot x$ for each $a\in\mathbf{X}_{e}=A$ and $x\in\mathbf{X}_{p}$. By the Hewitt–Cohen–Blanchard factorisation theorem, each $M_{p,e}$ is automatically an $A$-bimodule isomorphism. On the other hand, the maps $M_{e,p}$ need not be isomorphisms, since we do not require that each $\mathbf{X}_{p}$ is (left) nondegenerate (i.e. $M_{e,p}$ need not be surjective).

Given $p\in P\setminus\{e\}$ and $q\in P$, the $A$-bimodule isomorphism $M_{p,q}:\mathbf{X}_{p}\otimes_{A}\mathbf{X}_{q}\rightarrow\mathbf{X}_{pq}$ enables us to define a $*$-homomorphism $\iota_{p}^{pq}:\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)\rightarrow\mathcal{L}_{A}\left(\mathbf{X}_{pq}\right)$ by

[TABLE]

for each $S\in\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)$. Equivalently, the $*$-homomorphism $\iota_{p}^{pq}$ is characterised by the formula $\iota_{p}^{pq}(S)(xy)=(Sx)y$ for each $S\in\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)$, $x\in\mathbf{X}_{p}$, and $y\in\mathbf{X}_{q}$. Since $\mathbf{X}_{e}\otimes_{A}\mathbf{X}_{q}$ need not in general be isomorphic to $\mathbf{X}_{q}$, we cannot always define a map from $\mathcal{L}_{A}\left(\mathbf{X}_{e}\right)$ to $\mathcal{L}_{A}\left(\mathbf{X}_{q}\right)$ using the above procedure. However, as $\mathcal{K}_{A}\left(\mathbf{X}_{e}\right)=\mathcal{K}_{A}\left({}_{A}A_{A}\right)\cong A$, we can define $\iota_{e}^{q}:\mathcal{K}_{A}\left(\mathbf{X}_{e}\right)\rightarrow\mathcal{L}_{A}\left(\mathbf{X}_{q}\right)$ by $\iota_{e}^{q}(a):=\phi_{q}(a)$. For notational purposes, we define $\iota_{p}^{r}:\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)\rightarrow\mathcal{L}_{A}\left(\mathbf{X}_{r}\right)$ to be the zero map whenever $p,r\in P$ and $r\neq pq$ for all $q\in P$.

We are primarily interested in situations where the underlying semigroup possesses some additional order structure. In particular we focus on the quasi-lattice ordered groups introduced by Nica [13]. A quasi-lattice ordered group $(G,P)$ consists of a group $G$ and a subsemigroup $P$ of $G$ such that $P\cap P^{-1}=\{e\}$, and with respect to the partial order on $G$ induced by $p\leq q\Leftrightarrow p^{-1}q\in P$, any two elements $p,q\in G$ which have a common upper bound in $P$ have a least common upper bound in $P$. It is straightforward to show that if two elements in $G$ have a least common upper bound in $P$, then this least common upper bound is unique. We write $p\vee q$ for the least common upper bound of $p,q\in G$ if it exists. For $p,q\in G$, we write $p\vee q=\infty$ if $p$ and $q$ have no common upper bound in $P$, and $p\vee q<\infty$ otherwise. We can also extend the notion of least upper bounds in $(G,P)$ from pairs of elements in $P$ to finite subsets of $P$. We define $\bigvee\emptyset:=e$, $\bigvee\{p\}:=p$ for any $p\in P$, and for any $n\geq 2$ and $C:=\{p_{1},\ldots p_{n}\}\subseteq P$ we define $\bigvee C:=p_{1}\vee\cdots\vee p_{n}$ (since $P\cap P^{-1}=\{e\}$, the relation $\leq$ is antisymmetric, and so this is well-defined).

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a product system over $P$. We say that $\mathbf{X}$ is compactly aligned if, whenever $S\in\mathcal{K}_{A}(\mathbf{X}_{p})$ and $T\in\mathcal{K}_{A}(\mathbf{X}_{p})$ for some $p,q\in P$ with $p\vee q<\infty$, we have $\iota_{p}^{p\vee q}(S)\iota_{q}^{p\vee q}(T)\in\mathcal{K}_{A}(\mathbf{X}_{p\vee q})$. Note that this condition does not imply that either $\iota_{p}^{p\vee q}(S)$ or $\iota_{q}^{p\vee q}(T)$ is compact.

2.3. Representations of compactly aligned product systems, Nica covariance, and the Nica–Toeplitz algebra

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. A representation of $\mathbf{X}$ in a $C^{*}$-algebra $B$ is a map $\psi:\mathbf{X}\rightarrow B$ such that:

1. (T1)

each $\psi_{p}:=\psi|_{\mathbf{X}_{p}}$ is a linear map, and $\psi_{e}$ is a $C^{*}$-homomorphism; 2. (T2)

$\psi_{p}(x)\psi_{q}(y)=\psi_{pq}(xy)$ for all $p,q\in P$ and $x\in\mathbf{X}_{p}$, $y\in\mathbf{X}_{q}$; 3. (T3)

$\psi_{p}(x)^{*}\psi_{p}(y)=\psi_{e}(\langle x,y\rangle_{A}^{p})$ for all $p\in P$ and $x,y\in\mathbf{X}_{p}$.

It follows from (T1) and (T3) that a representation $\psi$ is always norm-decreasing, and isometric if and only $\psi_{e}$ is injective. Proposition 8.11 of [16] shows that for each $p\in P$, there exists a $*$-homomorphism $\psi^{(p)}:\mathcal{K}_{A}\left(\mathbf{X}_{p}\right)\rightarrow B$ such that $\psi^{(p)}\left(\Theta_{x,y}\right)=\psi_{p}(x)\psi_{p}(y)^{*}$ for all $x,y\in\mathbf{X}_{p}$.

We say that a representation $\psi:\mathbf{X}\rightarrow B$ is Nica covariant if, for any $p,q\in P$ and $S\in\mathcal{K}_{A}(\mathbf{X}_{p})$, $T\in\mathcal{K}_{A}(\mathbf{X}_{q})$, we have

[TABLE]

It follows from an application of the Hewitt–Cohen–Blanchard factorisation theorem that for any $p,q\in P$, we have

[TABLE]

Associated to each product system there exists a canonical Nica covariant representation called the Fock representation. We let $\mathcal{F}_{\mathbf{X}}:=\bigoplus_{p\in P}\mathbf{X}_{p}$ denote the space of sequences $(x_{p})_{p\in P}$ such that $x_{p}\in\mathbf{X}_{p}$ for each $p\in P$ and $\sum_{p\in P}\langle x_{p},x_{p}\rangle_{A}^{p}$ converges in $A$. By [11, Proposition 1.1] there exists a well defined $A$-valued inner product on $\mathcal{F}_{\mathbf{X}}$ such that $\left\langle(x_{p})_{p\in P},(y_{p})_{p\in P}\right\rangle_{A}=\sum_{p\in P}\langle x_{p},y_{p}\rangle_{A}^{p}$, and that $\mathcal{F}_{\mathbf{X}}$ is complete with respect to the induced norm. Letting $A$ act pointwise from the left and right gives $\mathcal{F}_{\mathbf{X}}$ the structure of a Hilbert $A$-bimodule, which we call the Fock space of $\mathbf{X}$. Lemma 5.3 of [6] shows that there exists an isometric Nica covariant representation $l:\mathbf{X}\rightarrow\mathcal{L}_{A}(\mathcal{F}_{\mathbf{X}})$ such that $l_{p}(x)(y_{q})_{q\in P}=\left(xy_{q}\right)_{q\in P}$ for each $p\in P$, $x\in\mathbf{X}_{p}$, and $(y_{q})_{q\in P}\in\mathcal{F}_{\mathbf{X}}$. We call $l$ the Fock representation of $\mathbf{X}$.

Using [12, Theorem 2.10] it can be shown that there exists a $C^{*}$-algebra $\mathcal{NT}_{\mathbf{X}}$, which we call the Nica–Toeplitz algebra of $\mathbf{X}$, and a Nica covariant representation $i_{\mathbf{X}}:\mathbf{X}\rightarrow\mathcal{NT}_{\mathbf{X}}$, that are universal in the following sense:

1. (i)

$\mathcal{NT}_{\mathbf{X}}$ is generated by the image of $i_{\mathbf{X}}$; 2. (ii)

if $\psi:\mathbf{X}\rightarrow B$ is any other Nica covariant representation of $\mathbf{X}$, then there exists a $*$-homomorphism $\psi_{*}:\mathcal{NT}_{\mathbf{X}}\rightarrow B$ such that $\psi_{*}\circ i_{\mathbf{X}}=\psi$.

Since $i_{\mathbf{X}}$ generates $\mathcal{NT}_{\mathbf{X}}$, it follows that $\mathcal{NT}_{\mathbf{X}}=\overline{\mathrm{span}}\left\{i_{\mathbf{X}}(x)i_{\mathbf{X}}(y)^{*}:x,y\in\mathbf{X}\right\}$.

Proposition 4.7 of [6] shows that there exists a coaction $\delta_{\mathbf{X}}:\mathcal{NT}_{\mathbf{X}}\rightarrow\mathcal{NT}_{\mathbf{X}}\otimes C^{*}(G)$ (we use an unadorned $\otimes$ to denote the minimal tensor product of $C^{*}$-algebras), which we call the canonical gauge coaction, such that $\delta_{\mathbf{X}}(i_{\mathbf{X}_{p}}(x))=i_{\mathbf{X}_{p}}(x)\otimes i_{G}(p)$ for each $p\in P$ and $x\in\mathbf{X}_{p}$. For those readers interested in learning more about coactions in general, we suggest [4, Appendix A].

Lemma 1.3 of [15] shows that there exists a conditional expectation $E_{\delta_{\mathbf{X}}}$ of $\mathcal{NT}_{\mathbf{X}}$ onto the generalised fixed-point algebra $\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}}:=\{b\in\mathcal{NT}_{\mathbf{X}}:\delta_{\mathbf{X}}(b)=b\otimes i_{G}(e)\}$ defined by $E_{\delta_{\mathbf{X}}}:=(\mathrm{id}_{\mathcal{NT}_{\mathbf{X}}}\otimes\rho)\circ\delta_{\mathbf{X}}$, where $\rho:C^{*}(G)\rightarrow{\mathbb{C}}$ is the canonical trace. It can be shown that $\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}}=\overline{\mathrm{span}}\{i_{\mathbf{X}_{p}}(\mathbf{X}_{p})i_{\mathbf{X}_{p}}(\mathbf{X}_{p})^{*}:p\in P\}$, and, for any $p,q\in P$, $x\in\mathbf{X}_{p}$, $y\in\mathbf{X}_{q}$, we have $E_{\delta_{\mathbf{X}}}\left(i_{\mathbf{X}_{p}}(x)i_{\mathbf{X}_{q}}(y)^{*}\right)=\delta_{p,q}i_{\mathbf{X}_{p}}(x)i_{\mathbf{X}_{q}}(y)^{*}$. We are particularly interested in the situation where the expectation $E_{\delta_{\mathbf{X}}}$ is faithful on positive elements, i.e. $E_{\delta_{\mathbf{X}}}(b^{*}b)=0\Rightarrow b=0$ for any $b\in\mathcal{NT}_{\mathbf{X}}$. Inspired by [6, Definition 7.1], we say that a compactly aligned product system $\mathbf{X}$ is amenable if $E_{\delta_{\mathbf{X}}}$ is faithful on positive elements. The argument of [10, Lemma 6.5] shows that if $G$ is an amenable group, then $\mathbf{X}$ is an amenable product system.

3. A uniqueness theorem for Nica–Toeplitz algebras

Firstly, we fix some notation.

Definition 3.1**.**

Let $(G,P)$ be a quasi-lattice ordered group, $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$, and $\psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H})$ a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. We define a collection $\{P_{p}^{\psi}:p\in P\}$ of projections in $\mathcal{B}(\mathcal{H})$ by $P_{e}^{\psi}:=\mathrm{id}_{\mathcal{H}}$ and $P_{p}^{\psi}:=\mathrm{proj}_{\overline{\psi_{p}(\mathbf{X}_{p})\mathcal{H}}}$ for each $p\in P\setminus\{e\}$. We also set $P_{\infty}^{\psi}:=0$.

The purpose of this article is to prove the following result:

Theorem 3.2**.**

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Suppose $\psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H})$ is a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$.

1. (i)

If the product system $\mathbf{X}$ is amenable, and, for any finite set $K\subseteq P\setminus\{e\}$, the representation

[TABLE]

is faithful, then the induced $*$-homomorphism $\psi_{*}:\mathcal{NT}_{\mathbf{X}}\rightarrow\mathcal{B}(\mathcal{H})$ is faithful. 2. (ii)

If $\psi_{*}$ is faithful and $\phi_{p}(A)\subseteq\mathcal{K}_{A}(\mathbf{X}_{p})$ for each $p\in P$, then the representation

[TABLE]

is faithful for any finite set $K\subseteq P\setminus\{e\}$.

The main step in the proof of the uniqueness theorem is to show that the expectation $E_{\delta_{\mathbf{X}}}$ is also implemented spatially, i.e. there is a compatible expectation $E_{\psi}$ of $\psi_{*}(\mathcal{NT}_{\mathbf{X}})$ onto $\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$. To get this compatible expectation we need to be able to calculate the norms of elements of $\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$. To do this we will make use of the following well-known fact about operators on Hilbert spaces: if $P_{1},\ldots,P_{n}\in\mathcal{B}(\mathcal{H})$ are mutually orthogonal projections that commute with $T\in\mathcal{B}(\mathcal{H})$ and satisfy $\sum_{i=1}^{n}P_{i}=\mathrm{id}_{\mathcal{H}}$, then $\left\|T\right\|_{\mathcal{B}(\mathcal{H})}=\max_{1\leq i\leq n}\left\|P_{i}T\right\|_{\mathcal{B}(\mathcal{H})}$.

We now work towards showing that there exists a collection of mutually orthogonal projections in $\mathcal{B}(\mathcal{H})$ that decompose the identity and commute with everything in $\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$.

We begin by showing that the $*$-homomorphism $\psi^{(p)}:\mathcal{K}_{A}(\mathbf{X}_{p})\rightarrow\mathcal{B}(\mathcal{H})$ has a canonical extension to all of $\mathcal{L}_{A}(\mathbf{X}_{p})$ (for each $p\in P$), and establish some properties of this extension.

Proposition 3.3**.**

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. Then

1. (i)

For each $p\in P$, there exists a representation $\rho_{p}^{\psi}:\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)\rightarrow\mathcal{B}(\mathcal{H})$ such that for each $S\in\mathcal{L}_{A}(\mathbf{X}_{p})$,

[TABLE]

and $\rho_{p}^{\psi}(S)$ is zero on $\left(\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{H}\right)^{\perp}$. 2. (ii)

$\rho_{p}^{\psi}|_{\mathcal{K}_{A}\left(\mathbf{X}_{p}\right)}=\psi^{(p)}$. 3. (iii)

For any $q\in P$ and $a\in A\cong\mathcal{K}_{A}(\mathbf{X}_{e})$, we have $\rho_{q}^{\psi}\left(\iota_{e}^{q}(a)\right)=\rho_{e}^{\psi}(a)P_{q}^{\psi}$. Furthermore, if $p\in P\setminus\{e\}$, then $\rho_{pq}^{\psi}\left(\iota_{p}^{pq}(S)\right)=\rho_{p}^{\psi}(S)P_{pq}^{\psi}$ for any and $S\in\mathcal{L}_{A}(\mathbf{X}_{p})$. 4. (iv)

If $\mathcal{K}\subseteq\mathcal{H}$ is a $\psi_{e}$-invariant subspace of $\mathcal{H}$, then the subspace $\mathcal{M}:=\overline{\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{K}}$ is $\rho_{p}^{\psi}$-invariant. Furthermore, if $\psi_{e}|_{\mathcal{K}}$ is faithful, then $\rho_{p}^{\psi}|_{\mathcal{M}}$ is also faithful.

Proof.

Observe that for any $p\in P$ and $x,y\in\mathbf{X}_{p}$ and $h,k\in\mathcal{H}$, we have

[TABLE]

Thus, there exists a linear isometry $U:\mathbf{X}_{p}\otimes_{A}\mathcal{H}\rightarrow\mathcal{H}$ such that $U\left(x\otimes_{A}h\right)=\psi_{p}(x)h$ for each $x\in\mathbf{X}_{p}$ and $h\in\mathcal{H}$. Equation (3.1) shows that $U^{*}\left(\psi_{p}(x)h\right)=x\otimes_{A}h$ for each $x\in\mathbf{X}_{p}$ and $h\in\mathcal{H}$. We claim that $U^{*}|_{\left(\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{H}\right)^{\perp}}=0$. To see this, observe that for any $f\in\left(\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{H}\right)^{\perp}$ and $y\in\mathbf{X}_{p}$, $h\in\mathcal{H}$ we have

[TABLE]

and hence $U^{*}(f)=0$. Since $\mathbf{X}_{p}\otimes_{A}\mathcal{H}=\left(\mathbf{X}_{p}\cdot A\right)\otimes_{A}\mathcal{H}=\mathbf{X}_{p}\otimes_{A}\overline{\psi_{e}(A)\mathcal{H}},$ we may assume that the representation $\psi_{e}$ is nondegenerate without loss of generality. With this in mind, define $\rho_{p}^{\psi}:\mathcal{L}_{A}(\mathbf{X}_{p})\rightarrow\mathcal{B}(\mathcal{H})$ by

[TABLE]

for each $S\in\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)$. Thus, for each $S\in\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)$, the restriction $\rho_{p}^{\psi}(S)|_{\left(\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{H}\right)^{\perp}}$ is zero, whilst for any $x\in\mathbf{X}_{p}$ and $h\in\mathcal{H}$ we have

[TABLE]

This completes the proof of part (i).

Since both $\psi^{(p)}$ and $\rho_{p}^{\psi}$ are $*$-homomorphisms, to prove (ii) it suffices to show that $\psi^{(p)}$ and $\rho_{p}^{\psi}$ agree on rank-one operators. Fix $x,y\in\mathbf{X}_{p}$. Firstly, we check that $\psi^{(p)}\left(\Theta_{x,y}\right)$ and $\rho_{p}^{\psi}\left(\Theta_{x,y}\right)$ agree on $\overline{\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{H}}$. For any $z\in\mathbf{X}_{p}$ and $h\in\mathcal{H}$, we have

[TABLE]

Since both $\psi^{(p)}\left(\Theta_{x,y}\right)$ and $\rho_{p}^{\psi}\left(\Theta_{x,y}\right)$ are linear and continuous, we conclude that they agree on $\overline{\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{H}}$. It remains to check that $\psi^{(p)}\left(\Theta_{x,y}\right)$ and $\rho_{p}^{\psi}\left(\Theta_{x,y}\right)$ agree on the orthogonal complement $\left(\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{H}\right)^{\perp}$. Making use of part (i), we see that the restriction $\rho_{p}^{\psi}\left(\Theta_{x,y}\right)|_{\left(\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{H}\right)^{\perp}}=0$. Since

[TABLE]

for any $h\in\left(\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{H}\right)^{\perp}$ and $k\in\mathcal{H}$, we conclude that $\psi^{(p)}(\Theta_{x,y})|_{\left(\psi_{p}\left(\mathbf{X}_{p}\right)\mathcal{H}\right)^{\perp}}=0$ as well. This completes the proof of part (ii).

We now prove part (iii). Let $q\in P$ and $a\in A$. If $q=e$, then $P_{q}^{\psi}=\mathrm{id}_{\mathcal{H}}$, and so

[TABLE]

On the other hand, if $q\neq e$, then $P_{q}^{\psi}=\mathrm{proj}_{\overline{\psi_{q}\left(\mathbf{X}_{q}\right)\mathcal{H}}}$. Hence, both $\rho_{e}^{\psi}(a)P_{q}$ and $\rho_{e}^{\psi}(\iota_{e}^{q}(a))$ are zero on $\left(\psi_{q}\left(\mathbf{X}_{q}\right)\mathcal{H}\right)^{\perp}$. Since $\rho_{q}^{\psi}\left(\iota_{e}^{q}(a)\right)$ and $\rho_{e}^{\psi}(a)P_{q}^{\psi}$ are linear and continuous, whilst

[TABLE]

for any $x\in\mathbf{X}_{q}$ and $h\in\mathcal{H}$, we see that $\rho_{q}^{\psi}\left(\iota_{e}^{q}(a)\right)$ and $\rho_{e}^{\psi}(a)P_{q}^{\psi}$ also agree on $\overline{\psi_{q}\left(\mathbf{X}_{q}\right)\mathcal{H}}$. Thus, $\rho_{q}^{\psi}\left(\iota_{e}^{q}(a)\right)=\rho_{e}^{\psi}(a)P_{q}^{\psi}$.

Now fix $p\in P\setminus\{e\}$ and $S\in\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)$. Since $pq\neq e$, both $\rho_{pq}^{\psi}\left(\iota_{p}^{pq}(S)\right)$ and $\rho_{p}^{\psi}(S)P_{pq}^{\psi}$ are zero on the orthogonal complement $\left(\psi_{pq}\left(\mathbf{X}_{pq}\right)\mathcal{H}\right)^{\perp}$. Observe that for any $x\in\mathbf{X}_{p}$, $y\in\mathbf{X}_{q}$, and $h\in\mathcal{H}$, we have

[TABLE]

which by part (i) is the same as

[TABLE]

Since $\overline{\psi_{pq}\left(\mathbf{X}_{pq}\right)\mathcal{H}}=\overline{\psi_{p}\left(\mathbf{X}_{p}\right)\psi_{q}\left(\mathbf{X}_{q}\right)\mathcal{H}}$, whilst $\rho_{pq}^{\psi}\left(\iota_{p}^{pq}(S)\right)$ and $\rho_{p}^{\psi}(S)P_{pq}^{\psi}$ are linear and continuous, we conclude that $\rho_{pq}^{\psi}\left(\iota_{p}^{pq}(S)\right)$ and $\rho_{p}^{\psi}(S)P_{pq}^{\psi}$ are also equal on $\overline{\psi_{pq}\left(\mathbf{X}_{pq}\right)\mathcal{H}}$.

Finally, we prove part (iv). Firstly, observe that the subspace $\mathcal{M}$ is $\rho_{p}^{\psi}$-invariant, since $\rho_{p}^{\psi}(S)\left(\psi_{p}(x)k\right)=\psi_{p}(Sx)k\in\mathcal{M}$ for any $S\in\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)$, $x\in\mathbf{X}_{p}$, and $k\in\mathcal{K}$. Now suppose that $\psi_{e}|_{\mathcal{K}}$ is faithful. Since $\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)$ acts faithfully on $\mathbf{X}_{p}$, the induced representation $\mathbf{X}_{p}\text{-}\mathrm{Ind}_{A}^{\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)}\left(\psi_{e}|_{\mathcal{K}}\right):\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)\rightarrow\mathcal{B}\left(\mathbf{X}_{p}\otimes_{A}\mathcal{K}\right)$ is faithful by [17, Corollary 2.74]. Since $U$ implements a unitary equivalence between $\mathbf{X}_{p}\text{-}\mathrm{Ind}_{A}^{\mathcal{L}_{A}\left(\mathbf{X}_{p}\right)}\left(\psi_{e}|_{\mathcal{K}}\right)$ and $\rho_{p}^{\psi}|_{\mathcal{M}}$, and unitary equivalence preserves the faithfulness of representations, we conclude that $\rho_{p}^{\psi}|_{\mathcal{M}}$ is faithful. ∎

We now show what a product of projections from the collection $\{P_{p}^{\psi}:p\in P\}$ looks like.

Proposition 3.4**.**

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. Then for each $p,q\in P$, we have

[TABLE]

In particular, the projections $\{P_{p}^{\psi}:p\in P\}$ commute.

Proof.

Firstly, observe that part (i) of Proposition 3.3 implies that $P_{p}^{\psi}=\rho_{p}^{\psi}\left(\mathrm{id}_{\mathbf{X}_{p}}\right)$ for any $p\in P\setminus\{e\}$.

Next, we show that if $p\in P$ and $(e_{i})_{i\in I}$ is the canonical approximate identity for the $C^{*}$-algebra $\mathcal{K}_{A}(\mathbf{X}_{p})$, then

1. (i)

$\lim_{i\in I}(e_{i}x)=x$ for each $x\in\mathbf{X}_{p}$; 2. (ii)

\lim_{i\in I}\big{(}\rho_{p}^{\psi}(e_{i})\big{)}=\rho_{p}^{\psi}\left(\mathrm{id}_{\mathbf{X}_{p}}\right) (converging in the strong operator topology).

To see (i), fix $x\in\mathbf{X}_{p}$ and $\varepsilon>0$. Choose $x^{\prime}\in\mathbf{X}_{p}$ so that $x=x^{\prime}\cdot\langle x^{\prime},x^{\prime}\rangle_{A}$ by the Hewitt–Cohen–Blanchard factorisation theorem. Choose $i\in I$ such that for all $j\geq i$,

[TABLE]

Thus, for all $j\geq i$, we have

[TABLE]

Since $\varepsilon>0$ was arbitrary, we conclude that $\lim_{i\in I}(e_{i}x)=x$ for each $x\in\mathbf{X}_{p}$.

We now move on to proving (ii). Fix $h\in\mathcal{H}$ and $\varepsilon>0$. If $h\in\left(\psi_{p}(\mathbf{X}_{p})\mathcal{H}\right)^{\perp}$, then

[TABLE]

for each $i\in I$. Thus, \lim_{i\in I}\big{\|}\rho_{p}^{\psi}(e_{i})h-\rho_{p}^{\psi}\left(\mathrm{id}_{\mathbf{X}_{p}}\right)h\big{\|}_{\mathcal{H}}=0. On the other hand, if $h\in\overline{\psi_{p}(\mathbf{X}_{p})\mathcal{H}}$, then we can choose $x_{1},\ldots,x_{n}\in\mathbf{X}_{p}$ and $h_{1},\ldots,h_{n}\in\mathcal{H}$ such that

[TABLE]

Since $\left\|e_{i}\right\|_{\mathcal{L}_{A}(\mathbf{X}_{p})}\leq 1$ for each $i\in I$ and $\rho_{p}^{\psi}$ is norm-decreasing, we see that

[TABLE]

By (i), for each $1\leq i\leq n$, we can choose $j_{i}\in I$ such that whenever $k\geq j_{i}$,

[TABLE]

As $I$ is directed, we can choose $m\in I$ such that $m\geq j_{i}$ for each $1\leq i\leq n$. Since $\psi_{p}$ is norm-decreasing, we see that for any $k\geq m$,

[TABLE]

Thus, for each $k\geq m$,

[TABLE]

Since $\varepsilon>0$ was arbitrary, we conclude that \lim_{i\in I}\big{\|}\rho_{p}^{\psi}(e_{i})h-\rho_{p}^{\psi}\left(\mathrm{id}_{\mathbf{X}_{p}}\right)h\big{\|}_{\mathcal{H}}=0 for each $h\in\mathcal{H}$. Thus, $\lim_{i\in I}\rho_{p}^{\psi}(e_{i})=\rho_{p}^{\psi}\left(\mathrm{id}_{\mathbf{X}_{p}}\right)$ in the strong operator topology.

Finally, we are ready to prove that $P_{p}^{\psi}P_{q}^{\psi}=P_{p\vee q}^{\psi}$ for every $p,q\in P$. Since $P_{e}^{\psi}=\mathrm{id}_{\mathcal{H}}$, the result is trivial when $p=e$ or $q=e$. Thus, we may as well suppose that $p,q\neq e$. Let $(e_{i})_{i\in I}$ and $(f_{j})_{j\in J}$ be the canonical approximate identities for $\mathcal{K}_{A}(\mathbf{X}_{p})$ and $\mathcal{K}_{A}(\mathbf{X}_{q})$ respectively. Then for any $i\in I$ and $j\in J$, Proposition 3.3 and the Nica covariance of $\psi$ tell us that

[TABLE]

Hence, by (ii), we have

[TABLE]

Thus, $P_{p}^{\psi}P_{q}^{\psi}=P_{p\vee q}^{\psi}$ for each $p,q\in P$. ∎

Our aim is to use the projections defined in Definition 3.1 to construct a collection of mutually orthogonal projections in $\mathcal{B}(\mathcal{H})$ that decompose the identity and commute with everything in $\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$.

Definition 3.5**.**

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. Let $F$ be a finite subset of $P$. For each $C\subseteq F$, define

[TABLE]

where, by convention, the product over the empty set is $\mathrm{id}_{\mathcal{H}}$.

Whilst we have defined the projections $Q_{C,F}^{\psi}$, for every subset $C$ of $F$, we are particularly interested in the projections corresponding to so called initial segments of $F$.

Definition 3.6**.**

Let $(G,P)$ be a quasi-lattice ordered group. Let $F\subseteq P$ be a finite set. A subset $C\subseteq F$ is said to be an initial segment of $F$ if $\bigvee C<\infty$ and $C=\left\{t\in F:t\leq\bigvee C\right\}$.

The next result shows how the projections \{Q_{C,F}^{\psi}:\text{C$is an initial segment of$F}\} and $\{P_{p}^{\psi}:p\in P\}$ interact with the operators $\{\psi_{p}(x):p\in P,\,x\in\mathbf{X}_{p}\}$.

Lemma 3.7**.**

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$.

1. (i)

Let $p,q\in P$ and $x\in\mathbf{X}_{p}$. Then

[TABLE] 2. (ii)

If $F\subseteq P$ is finite and $p\in F$, then

[TABLE]

for any initial segment $C$ of $F$.

Proof.

Fix $p,q\in P$ and $x\in\mathbf{X}_{p}$. If $p\vee q=\infty$, then $\psi_{q}(y)^{*}\psi_{p}(x)=0$ for any $y\in\mathbf{X}_{q}$. Hence, for any $h,g\in\mathcal{H}$, it follows that

[TABLE]

Thus, $\psi_{p}(\mathbf{X}_{p})\mathcal{H}\subseteq\left(\psi_{q}(\mathbf{X}_{q})\mathcal{H}\right)^{\perp}$, and so $P_{q}^{\psi}\psi_{p}(x)=0$.

Now suppose that $p\vee q<\infty$. If $p=p\vee q$, then $p\geq q$, and so

[TABLE]

If $p\neq p\vee q$, then for any $y\in\mathbf{X}_{p^{-1}(p\vee q)}$ and $h\in\mathcal{H}$, we have

[TABLE]

Consequently, $\psi_{p}(x)P_{p^{-1}(p\vee q)}^{\psi}$ and $P_{q}^{\psi}\psi_{p}(x)$ agree on $\overline{\psi_{p^{-1}(p\vee q)}(\mathbf{X}_{p^{-1}(p\vee q)})\mathcal{H}}$. It remains to check that they agree on the orthogonal complement $\left(\psi_{p^{-1}(p\vee q)}(\mathbf{X}_{p^{-1}(p\vee q)})\mathcal{H}\right)^{\perp}$. Let $f\in\left(\psi_{p^{-1}(p\vee q)}(\mathbf{X}_{p^{-1}(p\vee q)})\mathcal{H}\right)^{\perp}$. We need to show that $P_{q}^{\psi}\psi_{p}(x)f=0$. It suffices to show that $\psi_{p}(x)f\in\left(\psi_{q}(\mathbf{X}_{q})\mathcal{H}\right)^{\perp}$: for any $y\in\mathbf{X}_{q}$ and $h\in\mathcal{H}$, we have

[TABLE]

This completes the proof of (i).

We now prove part (ii). Fix a finite set $F\subseteq P$ with $p\in F$. Let $C$ be an initial segment of $F$. If $p\leq\bigvee C$, then $p\vee\left(\bigvee C\right)=\bigvee C<\infty$. By part (i) it follows that

[TABLE]

On the other hand, suppose that $p\not\leq\bigvee C$. Since $p\in F$, this implies that $C\neq F$. Moreover, since $C$ is an initial segment of $F$, this forces $p\in F\setminus C$. Therefore,

[TABLE]

This completes the proof of part (ii). ∎

Using the previous result we can show that every element of $\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$ commutes with the projections $\{P_{p}^{\psi}:p\in P\}$ and $\{Q_{C,F}^{\psi}:C\subseteq F\}$.

Proposition 3.8**.**

For any $q\in P$, the projection $P_{q}^{\psi}$ commutes with every element of $\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$. In particular, if $F\subseteq P$ is finite and $C\subseteq F$, then $Q_{C,F}^{\psi}$ commutes with every element of $\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$.

Proof.

Since $\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})=\overline{\mathrm{span}}\{\psi_{p}(\mathbf{X}_{p})\psi_{p}(\mathbf{X}_{p})^{*}\}$, it suffices to show that

[TABLE]

for each $x,y\in\mathbf{X}_{p}$. Via two applications of Lemma 3.7, we see that

[TABLE]

as required. ∎

We now show that the projections \{Q_{C,F}^{\psi}:\text{C$is an initial segment of$F}\} are mutually orthogonal and decompose the identity operator on $\mathcal{H}$.

Proposition 3.9**.**

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$. Let $F\subseteq P$ be finite. Then

1. (i)

if $C\subseteq F$ is not an initial segment of $F$, then $Q_{C,F}^{\psi}=0$; 2. (ii)

$\{Q_{C,F}^{\psi}:C\subseteq F\text{ is an initial segment of }F\}$* is a decomposition of the identity on $\mathcal{H}$ into mutually orthogonal projections.*

Proof.

Suppose $C\subseteq F$ is not an initial segment of $F$. If $\bigvee C=\infty$, then

[TABLE]

Alternatively, $\bigvee C<\infty$ and $C\neq\{t\in F:t\leq\bigvee C\}$. Thus, $C\neq F$. Choose $t\in F\setminus C$ with $t\leq\bigvee C$. Since $t\vee\left(\bigvee C\right)=\bigvee C$, we see that

[TABLE]

Thus, $Q_{C,F}^{\psi}=0$, which proves part (i).

We now prove part (ii). Since $Q_{C,F}^{\psi}=0$ whenever $C$ is not an initial segment of $F$, it suffices to show that $\{Q_{C,F}^{\psi}:C\subseteq F\}$ is a decomposition of the identity into mutually orthogonal projections. Firstly, we show orthogonality. Suppose $C,D\subseteq F$ are distinct. Without loss of generality, we may assume that $D\setminus C\neq\emptyset$. Thus, $C\neq F$ and we can choose $t\in D\setminus C$. Since $t\vee\left(\bigvee D\right)=\bigvee D$, we have

[TABLE]

It remains to check that $\sum_{C\subseteq F}Q_{C,F}^{\psi}=\mathrm{id}_{\mathcal{H}}$. To prove this, we will use induction on $|F|$. When $|F|=0$ we have

[TABLE]

Now let $n\geq 0$ and suppose that $\sum_{C\subseteq F}Q_{C,F}^{\psi}=\mathrm{id}_{\mathcal{H}}$ whenever $F\subseteq P$ and $|F|=n$. Fix $F^{\prime}\subseteq P$ with $|F^{\prime}|=n+1$. Then, for any $y\in F^{\prime}$, we have

[TABLE]

where the last equality follows from applying the inductive hypothesis to $F^{\prime}\setminus\{y\}$. ∎

Putting these results together we get an expression for the norm of an element in $\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$ that does not depend on the representation $\psi$.

Lemma 3.10**.**

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. If $Z:=\sum_{k}\psi_{p_{k}}(x_{k})\psi_{p_{k}}(y_{k})^{*}\in\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$ is a finite sum, then

[TABLE]

for any finite set $F\subseteq P$ containing each $p_{k}$. Furthermore, if the representation

[TABLE]

is faithful for any finite set $K\subseteq P\setminus\{e\}$, then

[TABLE]

Proof.

Let $F$ be a finite subset of $P$ containing each $p_{k}$. Since $Q_{C,F}^{\psi}$ commutes with $Z$ for each $C\subseteq F$ (by Proposition 3.8) and \{Q_{C,F}^{\psi}:\text{C$is an initial segment of$F}\} is an orthogonal decomposition of the identity, we have that

[TABLE]

However, for any initial segment $C$ of $F$, Lemma 3.7 shows that

[TABLE]

Using parts (ii) and (iii) of Proposition 3.3, this is equal to

[TABLE]

Now suppose that for any finite set $K\subseteq P\setminus\{e\}$, the representation

[TABLE]

is faithful. To complete the proof we will show that the representation

[TABLE]

is faithful, and hence

[TABLE]

Let $\mathcal{K}:=\prod_{\left\{t\in F\setminus C:t\vee\left(\bigvee C\right)<\infty\right\}}\left(\mathrm{id}_{\mathcal{H}}-P_{\left(\bigvee C\right)^{-1}\left(t\vee\left(\bigvee C\right)\right)}^{\psi}\right)\mathcal{H}$. Since $\psi_{e}(a)P_{p}^{\psi}=P_{p}^{\psi}\psi_{e}(a)$, for each $a\in A$ and $p\in P$, by Lemma 3.7, we see that $\mathcal{K}$ is a $\psi_{e}$-invariant subspace of $\mathcal{H}$. As $C$ is an initial segment of $F$, if $t\in F\setminus C$ with $t\vee\left(\bigvee C\right)<\infty$, then $t\not\leq\bigvee C$, and so $\left(\bigvee C\right)^{-1}\left(t\vee\left(\bigvee C\right)\right)\neq e$. Thus, $\psi_{e}|_{\mathcal{K}}$ is faithful. Therefore, by Proposition 3.3 it follows that $\mathcal{M}:=\overline{\psi_{\bigvee C}(\mathbf{X}_{\bigvee C})\mathcal{K}}$ is a $\rho_{\bigvee C}^{\psi}$-invariant subspace and $\rho_{\bigvee C}^{\psi}|_{\mathcal{M}}$ is faithful. To show that the map $\mathcal{L}_{A}\left(\mathbf{X}_{\bigvee C}\right)\ni T\mapsto Q_{C,F}^{\psi}\rho_{\bigvee C}^{\psi}(T)\in\mathcal{B}(\mathcal{H})$ is faithful, it remains to check that $\mathcal{M}\subseteq Q_{C,F}^{\psi}\mathcal{H}$. Lemma 3.7 tells us that for any $x\in\mathbf{X}_{\bigvee C}$, we have

[TABLE]

Therefore, $Q_{C,F}^{\psi}$ is the identity on

[TABLE]

and so $\mathcal{M}\subseteq Q_{C,F}^{\psi}\mathcal{H}$. ∎

Now that we have an expression for the norm of elements in $\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$, we are ready to show that the expectation $E_{\delta_{\mathbf{X}}}$ can be implemented spatially.

Proposition 3.11**.**

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. Suppose that for any finite set $K\subseteq P\setminus\{e\}$, the representation

[TABLE]

is faithful. Then

1. (i)

$\psi_{*}|_{\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}}}$* is faithful; and* 2. (ii)

there exists a linear map $E_{\psi}:\psi_{*}\left(\mathcal{NT}_{\mathbf{X}}\right)\rightarrow\psi_{*}(\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}})$ such that

[TABLE]

Proof.

Firstly, we prove that the restriction of $\psi_{*}$ to $\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}}$ is faithful. Fix a finite sum $Z:=\sum_{k}i_{\mathbf{X}_{p_{k}}}(x_{k})i_{\mathbf{X}_{p_{k}}}(y_{k})^{*}\in\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}}$. Let $\sigma:\mathcal{NT}_{\mathbf{X}}\rightarrow\mathcal{B}(\mathcal{H}^{\prime})$ be a faithful representation. Thus, $\sigma\circ i_{\mathbf{X}}$ is a Nica covariant representation of $\mathbf{X}$ on $\mathcal{H}^{\prime}$. For any finite set $F\subseteq P$ containing each $p_{k}$, two applications of Lemma 3.10 show that

[TABLE]

where we used the fact that each $Q_{C,F}^{\sigma\circ i_{\mathbf{X}}}$ is a projection and each $*$-homomorphism $\rho_{\bigvee C}^{\sigma\circ i_{\mathbf{X}}}$ is norm-decreasing. Thus, $\psi_{*}|_{\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}}}$ is faithful.

Next we prove part (ii). We first show that for any finite sum $\sum_{k}\psi_{p_{k}}(x_{k})\psi_{q_{k}}(y_{k})^{*}$, we have

[TABLE]

Let $F\subseteq P$ be the finite set consisting of each $p_{k}$ and $q_{k}$. By Lemma 3.10, there exists an initial segment $C$ of $F$ such that

[TABLE]

For each $s,t\in C$ with $s\neq t$ and $\left(s^{-1}\left(\bigvee C\right)\right)\vee\left(t^{-1}\left(\bigvee C\right)\right)<\infty$ define

[TABLE]

Observe that for any $s,t\in C$ with $s\neq t$ and $\left(s^{-1}\left(\bigvee C\right)\right)\vee\left(t^{-1}\left(\bigvee C\right)\right)<\infty$ we have

[TABLE]

and

[TABLE]

Thus, $\beta_{s,t}\geq\bigvee C$. Hence, $P_{\bigvee C}^{\psi}P^{\psi}_{\beta_{s,t}}=P^{\psi}_{\left(\bigvee C\right)\vee\beta_{s,t}}=P^{\psi}_{\beta_{s,t}}$, and we can define a projection

[TABLE]

We claim that

[TABLE]

Since $R_{C,F}^{\psi}$ commutes with $\sum_{k:p_{k}=q_{k}}\psi_{p_{k}}(x_{k})\psi_{p_{k}}(y_{k})^{*}$ by Proposition 3.8, it suffices to show that $R_{C,F}^{\psi}\psi_{p}(x)\psi_{q}(y)^{*}R_{C,F}^{\psi}$=0 whenever $x\in\mathbf{X}_{p}$, $y\in\mathbf{X}_{q}$, with $p,q\in F$ and $p\neq q$. Firstly, if $p\not\in C$ or $q\not\in C$, then $p\not\leq\bigvee C$ or $q\not\leq\bigvee C$ (since $C$ is an initial segment of $F$), and so by Lemma 3.7 we have $Q_{C,F}^{\psi}\psi_{p}(x)=0$ or $Q_{C,F}^{\psi}\psi_{q}(y)=0$. Consequently,

[TABLE]

Alternatively, if $p,q\leq\bigvee C$ and $\left(p^{-1}\left(\bigvee C\right)\right)\vee\left(q^{-1}\left(\bigvee C\right)\right)=\infty$, then

[TABLE]

Hence, Lemma 3.7 tells us that

[TABLE]

With this in mind, suppose that $p,q\in C$ and $\left(p^{-1}\left(\bigvee C\right)\right)\vee\left(q^{-1}\left(\bigvee C\right)\right)<\infty$. Since $p$ and $q$ are distinct, it follows that either $p^{-1}\left(\bigvee C\right)<\left(p^{-1}\left(\bigvee C\right)\right)\vee\left(q^{-1}\left(\bigvee C\right)\right)$ or $q^{-1}\left(\bigvee C\right)<\left(p^{-1}\left(\bigvee C\right)\right)\vee\left(q^{-1}\left(\bigvee C\right)\right)$. By taking adjoints, we may assume, without loss of generality, that $p^{-1}<\left(p^{-1}\left(\bigvee C\right)\right)\vee\left(q^{-1}\left(\bigvee C\right)\right)$. Therefore,

[TABLE]

Consequently, an application of Lemma 3.7 shows that

[TABLE]

Additionally, we claim that the representation

[TABLE]

is faithful. Let

[TABLE]

As $\psi_{e}(a)P_{q}^{\psi}=P_{q}^{\psi}\psi_{e}(a)$, for each $a\in A$ and $q\in P$, by Lemma 3.7, we see that $\mathcal{K}$ is a $\psi_{e}$-invariant subspace of $\mathcal{H}$. Since $C$ is an initial segment of $F$, if $p\in F\setminus C$ with $p\vee\left(\bigvee C\right)<\infty$, then $p\not\leq\bigvee C$, and so $\left(\bigvee C\right)^{-1}\left(p\vee\left(\bigvee C\right)\right)\neq e$. We also claim that for any $s,t\in C$ with $s\neq t$ and $\left(s^{-1}\left(\bigvee C\right)\right)\vee\left(t^{-1}\left(\bigvee C\right)\right)<\infty$, we have $\left(\bigvee C\right)^{-1}\beta_{s,t}\neq e$. Firstly, if $s^{-1}\left(\bigvee C\right)<\left(s^{-1}\left(\bigvee C\right)\right)\vee\left(t^{-1}\left(\bigvee C\right)\right)$, then

[TABLE]

On the other hand, if $s^{-1}\left(\bigvee C\right)=\left(s^{-1}\left(\bigvee C\right)\right)\vee\left(t^{-1}\left(\bigvee C\right)\right)$, then

[TABLE]

since $s\neq t$. Thus, $\psi_{e}|_{\mathcal{K}}$ is faithful. Hence, by Proposition 3.3 it follows that the subspace $\mathcal{M}:=\overline{\psi_{\bigvee C}(\mathbf{X}_{\bigvee C})\mathcal{K}}$ is $\rho_{\bigvee C}^{\psi}$-invariant and $\rho_{\bigvee C}^{\psi}|_{\mathcal{M}}$ is faithful. To see that the map $\mathcal{L}_{A}\left(\mathbf{X}_{\bigvee C}\right)\ni T\mapsto R_{C,F}^{\psi}\rho_{\bigvee C}^{\psi}(T)\in\mathcal{B}(\mathcal{H})$ is faithful, it remains to show that $\mathcal{M}\subseteq R_{C,F}^{\psi}\mathcal{H}$. Suppose $s,t\in C$ with $s\neq t$ and $\left(s^{-1}\left(\bigvee C\right)\right)\vee\left(t^{-1}\left(\bigvee C\right)\right)<\infty$. Since $\bigvee C\leq\beta_{s,t}$, Lemma 3.7 tells us that for any $x\in\mathbf{X}_{\bigvee C}$ we have

[TABLE]

Thus,

[TABLE]

Hence, $R_{C,F}^{\psi}$ is the identity on

[TABLE]

and so $\mathcal{M}\subseteq R_{C,F}^{\psi}\mathcal{H}$.

Putting all of this together, we see that

[TABLE]

Since the norm estimate (3.2) holds, the formula $\psi_{p}(x)\psi_{q}(y)^{*}\mapsto\delta_{p,q}\psi_{p}(x)\psi_{q}(y)^{*}$ extends to a map on $\psi_{*}(\mathcal{NT}_{\mathbf{X}})=\overline{\mathrm{span}}\left\{\psi_{p}(x)\psi_{q}(y)^{*}:p,q\in P,\ x\in\mathbf{X}_{p},\ y\in\mathbf{X}_{q}\right\}$ by linearity and continuity, which we denote by $E_{\psi}$. Furthermore, for any $p,q\in P$, $x\in\mathbf{X}_{p}$, $y\in\mathbf{X}_{q}$, we have

[TABLE]

Since $\mathcal{NT}_{\mathbf{X}}=\overline{\mathrm{span}}\left\{i_{\mathbf{X}_{p}}(x)i_{\mathbf{X}_{q}}(y)^{*}:p,q\in P,x\in\mathbf{X}_{p},y\in\mathbf{X}_{q}\right\}$, whilst the maps $E_{\psi}\circ\psi_{*}$ and $\psi_{*}\circ E_{\delta_{\mathbf{X}}}$ are linear and norm-decreasing, we conclude that $E_{\psi}\circ\psi_{*}=\psi_{*}\circ E_{\delta_{\mathbf{X}}}$. This completes the proof of part (ii). ∎

Finally, we prove the uniqueness theorem for Nica–Toeplitz algebras. We remind the reader that if $(G,P)$ is a quasi-lattice ordered group with $G$ amenable, then any compactly aligned product system $\mathbf{X}$ over $P$ is automatically amenable (in the sense that the expectation $E_{\delta_{\mathbf{X}}}$ is faithful on positive elements).

Proof of Theorem 3.2.

Suppose that the representation

[TABLE]

is faithful for any finite set $K\subseteq P\setminus\{e\}$. Let $b\in\mathcal{NT}_{\mathbf{X}}$ be such that $\psi_{*}(b)=0$. Thus,

[TABLE]

Since $\psi_{*}$ is faithful on $\mathcal{NT}_{\mathbf{X}}^{\delta_{\mathbf{X}}}=E_{\delta_{\mathbf{X}}}(\mathcal{NT}_{\mathbf{X}})$, we must have $E_{\delta_{\mathbf{X}}}(b^{*}b)=0$. As $E_{\delta_{\mathbf{X}}}$ is faithful on positive elements, we conclude that $b=0$. Hence, $\psi_{*}$ is faithful.

We now prove (ii). Suppose that $\psi_{*}$ is faithful and $\phi_{p}(A)\subseteq\mathcal{K}_{A}(\mathbf{X}_{p})$ for each $p\in P$. Fix a finite set $K\subseteq P\setminus\{e\}$. For each $a\in A$, define

[TABLE]

We claim that for any Nica covariant representation $\varphi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H}^{\prime})$ we have

[TABLE]

To see this, firstly observe that for any $t\in P$ and $a\in A$, we have

[TABLE]

Therefore,

[TABLE]

Hence, to prove that Equation (3.3) holds, it suffices to show that

[TABLE]

To prove this we use induction on $|K|$. When $|K|=0$ we have

[TABLE]

Now let $n\in{\mathbb{N}}$ and suppose we have equality whenever $K\subseteq P$ and $|K|=n$. Fix $K^{\prime}\subseteq P$ with $|K^{\prime}|=n+1$. Let $s\in K^{\prime}$. Then

[TABLE]

This proves that Equation (3.4) holds, and so Equation (3.3) follows.

Now let $\pi:A\rightarrow\mathcal{B}(\mathcal{H}^{\prime})$ be a faithful nondegenerate representation of $A$. Define $\Psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{F}_{\mathbf{X}}\otimes_{A}\mathcal{H}^{\prime})$ by

[TABLE]

where $l:\mathbf{X}\rightarrow\mathcal{L}_{A}(\mathcal{F}_{\mathbf{X}})$ is the Fock representation of $\mathbf{X}$. Since $l$ is a Nica covariant representation of $\mathbf{X}$ and $\mathcal{F}_{\mathbf{X}}\text{-}\mathrm{Ind}_{A}^{\mathcal{L}_{A}(\mathcal{F}_{\mathbf{X}})}\pi$ is a $*$-homomorphism, $\Psi$ is a Nica covariant representation of $\mathbf{X}$. We claim that the representation

[TABLE]

is faithful. To see this, suppose that $a\in A\setminus\{0\}$, so that $aa^{*}\neq 0$. As $\pi$ is faithful, we can find $h\in\mathcal{H}^{\prime}$ such that $\pi(aa^{*})h\neq 0$. For any $t\in P\setminus\{e\}$ we have

[TABLE]

Hence, it follows that $P_{t}^{\Psi}=\mathrm{proj}_{\overline{\Psi_{t}(\mathbf{X}t)\left(\mathcal{F}_{\mathbf{X}}\otimes_{A}\mathcal{H}^{\prime}\right)}}$ is zero on $A\otimes_{A}\mathcal{H}^{\prime}\subseteq\mathcal{F}_{\mathbf{X}}\otimes_{A}\mathcal{H}^{\prime}$. Thus, as $e\not\in K$ we see that

[TABLE]

which is nonzero because

[TABLE]

Therefore, $A\ni a\mapsto\Psi_{e}(a)\prod_{t\in K}\left(\mathrm{id}_{\mathcal{F}_{\mathbf{X}}\otimes_{A}\mathcal{H}^{\prime}}-P_{t}^{\Psi}\right)\in\mathcal{B}(\mathcal{F}_{\mathbf{X}}\otimes_{A}\mathcal{H}^{\prime})$ is faithful.

Putting all of this together, and using that $\psi_{*}$ is faithful at the penultimate equality, we see that for any $a\in A$,

[TABLE]

Hence, A\ni a\mapsto\psi_{e}(a)\prod_{t\in K}\big{(}\mathrm{id}_{\mathcal{H}}-P_{t}^{\psi}\big{)}\in\mathcal{B}(\mathcal{H}) is faithful. ∎

In practice, we are often interested in representations of product systems in more general $C^{*}$-algebras, rather than on Hilbert spaces. The following corollary shows that provided the coefficient algebra acts compactly on each fibre of the product system, we can still characterise the faithfulness of the induced representation.

Corollary 3.12**.**

Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ an amenable compactly aligned product system over $P$ with coefficient algebra $A$. Suppose that $A$ acts compactly on each $\mathbf{X}_{p}$. Let $\psi:\mathbf{X}\rightarrow B$ be a Nica covariant representation of $\mathbf{X}$ in a $C^{*}$-algebra $B$. Then the induced $*$-homomorphism $\psi_{*}:\mathcal{NT}_{\mathbf{X}}\rightarrow B$ is faithful if and only if for every $a\in A\setminus\{0\}$ and every finite set $K\subseteq P\setminus\{e\}$, we have

[TABLE]

Proof.

Fix a faithful representation $\pi:B\rightarrow\mathcal{B}(\mathcal{H})$. Then $\pi\circ\psi:\mathbf{X}\rightarrow\mathcal{B}(\mathcal{H})$ is a Nica covariant representation with induced representation $(\pi\circ\psi)_{*}=\pi\circ\psi_{*}$. If $a\in A$ and $t\in P\setminus\{e\}$, then Proposition 3.3 implies that

[TABLE]

Hence, for any $a\in A$ and any finite set $K\subseteq P\setminus\{e\}$, we have that

[TABLE]

and so the result follows from Theorem 3.2. ∎

4. Acknowledgements

The results in this article are from my PhD thesis. Thank you to my supervisors Adam Rennie and Aidan Sims at the University of Wollongong for their advice and encouragement during my PhD and during the writing of this article.