A proof of Boca's Theorem

TL;DR

This paper introduces a general method for extending unital completely positive maps to amalgamated free products of C*-algebras and provides a dilation theoretic proof of Boca's Theorem.

Contribution

It presents a novel approach to extending maps and offers a new proof of Boca's Theorem using dilation theory.

Findings

Extension method for unital completely positive maps

Dilation theoretic proof of Boca's Theorem

Applicable to amalgamated free products of C*-algebras

Abstract

We give a general method of extending unital completely positive maps to amalgamated free products of C*-algebras. As an application we give a dilation theoretic proof of Boca's Theorem.

Equations

$$\{{\psi }_i:A_i\to \mathcal{B}(H)\mid {\psi }_i{\epsilon }_i={\psi }_j{\epsilon }_j\text{ for }i,j\in I\}$$

$${\text{bigast}}_B{A}_i=B\oplus \underset{i_1\cdots i_n\in S}{\sum ^{\oplus }}(\ker E_{i_1}{\otimes }_B\cdots {\otimes }_B\ker E_{i_n})$$

$$S:=\{i_1\cdots i_n:n\ge 1,i_1,\dots ,i_n\in I,i_1\ne \cdots \ne i_n\}$$

$$\Phi (b)=\rho (b)\text{and}\Phi (a_{i_1}\ast \cdots \ast a_{i_n}):={\Phi }_{i_1}(a_{i_1})\cdots {\Phi }_{i_n}(a_{i_n})$$

$$\Vert x\Vert :=\sup \{\Vert \pi (x)\Vert :\text{ π is a ∗-rep-re-sent-ation of bigastBAi}\}.$$

$${\rho }_i\oplus {\pi }_i{\epsilon }_i={\rho }_j\oplus {\pi }_j{\epsilon }_j\text{ for all }i,j\in I.$$

$$s(i_1\cdots i_n):=i_n\text{and}|i_1\cdots i_n|:=n.$$

$$\underset{s(w)\ne i}{\sum ^{\oplus }}{H}_w\oplus H_{wi}=\underset{u\in S,u\ne i}{\sum ^{\oplus }}{H}_u=K\ominus H_i.$$

$${\pi }_i{\epsilon }_i=\underset{s(w)\ne i}{\sum ^{\oplus }}{\rho }_w\oplus {\rho }_{wi}=\underset{u\in S,u\ne i}{\sum ^{\oplus }}{\rho }_u.$$

$${\rho }_i\oplus {\pi }_i{\epsilon }_i={\rho }_j\oplus {\pi }_j{\epsilon }_j.$$

$${\tau }_i{\epsilon }_i$$

$$C_i=\{\begin{array}{ll}{A}_i& \text{ if }i\in J,\\ B& \text{ if }i\notin J.\end{array}$$

$${\text{bigast}}_B{C}_i=({\text{bigast}}_B^{i\in J}{A}_i){\text{bigast}}_B({\text{bigast}}_B^{i\notin J}B)=({\text{bigast}}_B^{i\in J}{A}_i){\text{bigast}}_{B}B=({\text{bigast}}_B^{i\in J}{A}_i).$$

$${\rho }_i\oplus {\pi }_i{\epsilon }_i={\rho }_j\oplus {\pi }_j{\epsilon }_j.$$

$${\tau }_i:={\sigma }_i\oplus {\pi }_i:A_i\to \mathcal{B}(H\oplus K)$$

$$\tau :{\check{\text{bigast}}}_B{A}_i\to \mathcal{B}(H\oplus K)\text{ with }\tau {|}_{A_i}={\tau }_i.$$

$${\rho }_i:=P_{M_i}{\sigma }_i{\epsilon }_i:B\to \mathcal{B}(M_i).$$

$$U_i{|}_H=I_H\text{and}{U}_i^{\ast }{\rho }_i(b)U_i={\rho }_{i_0}(b)\text{ for all }b\in B.$$

$${\Psi }_i(a)=U_i^{\ast }{P}_{M_i}{\sigma }_i(a)U_i\text{ for all }a\in A_i.$$

$$P_H{U}_i^{\ast }{P}_{M_i}{\sigma }_i(a)U_i{|}_H=P_H{\sigma }_i(a){|}_H={\Phi }_i(a).$$

$${\rho }_i\oplus {\pi }_i{\epsilon }_i={\rho }_j\oplus {\pi }_j{\epsilon }_j\text{ for all }i,j\in I$$

$${\pi }_j(a){|}_{H_w\oplus H_{wj}}={\pi }_{j,w}(a)=\left[\begin{array}{cc}0& \ast \\ \ast & \ast \end{array}\right]$$

$$\Phi (a_n\cdots a_1)={\Phi }_{j_n}(a_n)\cdots {\Phi }_{j_1}(a_1)$$

$$\tau_{i}:=\sigma_{i}\oplus\pi_{i}:A_{i}\to{\mathcal{B}}\big{(}H\oplus\operatornamewithlimits{\sum^{\oplus}}_{w\in S}H_{w}\big{)}$$

$${\tau }_i(a)h\in {\Phi }_i(a)h+H_i\text{ for all }i\in I,$$

$${\pi }_j(\ker E_j)H_w\subset H_{wj}\text{ when }j\in J\text{ and }s(w)\ne j.$$

$$\Phi (a_n\cdots a_1)=P_H{\tau }_{j_n}(a_n)\cdots {\tau }_{j_1}(a_1){|}_H.$$

$${\tau }_{j_n}(a_n)\cdots {\tau }_{j_1}(a_1)h\in {\Phi }_{j_n}(a_n)\cdots {\Phi }_{j_1}(a_1)h+\underset{s(w)=j_n}{\sum ^{\oplus }}{H}_w$$

$${\tau }_{j_n}(a_n)\cdots {\tau }_{j_1}(a_1)h$$

$$\subseteq {\Phi }_{j_n}(a_n)\cdots {\Phi }_{j_1}(a_1)h+H_{j_n}+\underset{s(w)=j_{n-1}}{\sum ^{\oplus }}{H}_{w{j}_n}$$

$$\subseteq {\Phi }_{j_n}(a_n)\cdots {\Phi }_{j_1}(a_1)h+\underset{s(w)=j_n}{\sum ^{\oplus }}{H}_w.$$

Full text

1. Introduction

The amalgamated free product of C*-algebras has become a standard construction in the subject. It is central in studying group C*-algebras, free probability, quantum information, dynamical systems on C*-algebras, and the Connes Embedding Conjecture via its connection to Kirchberg’s conjecture and also to Tsirelson’s conjectures. Boca’s result on completely positive maps on amalgamated free products has become a useful technical tool in this endeavour. A comprehensive survey would be too time consuming for this short note.

Avitzour [2] showed that states on two C*-algebras can be extended to a state on the free product of the two C*-algebras. Boca’s Theorem [4] shows that unital completely positive maps on two C*-algebras agreeing on a common C*-subalgebra can be extended to a unital completely positive map on their amalgamated free product. However his method requires some additional structure, and yields additional structure. Here we give a simple proof of the basic result that does not require these added hypotheses, and then we explain how the argument can be modified to yield Boca’s actual theorem.

To fix notation, let $\{A_{i}\}_{i\in I}$ be a family of unital C*-algebras with a common unital C*-subalgebra $B$, i.e., there are unital imbeddings $\varepsilon_{i}:B\to A_{i}$. The amalgamated free product $\check{\bigast}_{B}A_{i}$ is an appropriate completion of the $*$-algebraic amalgamated free product $\bigast_{B}A_{i}$. It is the universal C*-algebra generated by $\bigcup_{i}\varphi_{i}(A_{i})$, for imbeddings $\{\varphi_{i}\}_{i\in I}$ with $\varphi_{i}\varepsilon_{i}=\varphi_{j}\varepsilon_{j}$, such that: every family of $*$-representations

[TABLE]

lifts to a $*$-representation $\pi:\check{\bigast}_{B}A_{i}\to{\mathcal{B}}(H)$ such that $\psi_{i}=\pi\varphi_{i}$.

If we assume the existence of conditional expectations $E_{i}:A_{i}\to B$ for $i\in I$, then the $*$-algebraic amalgamated free product, as a linear space, takes the form

[TABLE]

where $\otimes_{B}$ denotes the bimodule tensor product, and

[TABLE]

is the set of non-empty words in the alphabet $I$ that contain no subwords of the form $ii$. See [2] or [6, Proof of Theorem 3.1]. It is natural to use (1.1) for extending linear maps canonically. Indeed let $\{\Phi_{i}:A_{i}\to{\mathcal{B}}(H)\}_{i\in I}$ be unital completely positive maps which restrict to a common $*$-representation $\rho$ of $B$. Then one can directly define a linear map $\Phi$ on $\bigast_{B}A_{i}$ given by

[TABLE]

for $b\in B$ and $a_{i_{k}}\in\ker E_{i_{k}}$ when $i_{1}\cdots i_{n}\in S$. Boca shows that $\Phi$ extends to a unital completely positive map of $\check{\bigast}_{B}A_{i}$. Applying this to $B={\mathbb{C}}$ yields the fact that unital completely positive maps of $A_{i}$ into a common Hilbert space extend to a unital completely positive map of $\check{\bigast}A_{i}$.

To achieve his results, Boca verifies matrix inequalities for elements in $\bigast_{B}A_{i}$. However this line of reasoning does not apply in the absence of expectations. Here we present a method that tackles this problem by exploiting the ideas of [7]. This produces an explicit Stinespring dilation of the desired completely positive map. Boca [5] produces an explicit Stinespring dilation by different methods when expectations are available. With additional care, our arguments also provide an alternative proof of Boca’s result.

Our strategy is to dilate the maps $\{\Phi_{i}\}_{i\in I}$ to $*$-representations $\{\pi_{i}:A_{i}\to{\mathcal{B}}(K)\}_{i\in I}$ on a common Hilbert space $K$ that agree on $B$. The universal property will then provide a $*$-representation $\pi$ of $\check{\bigast}_{B}A_{i}$. The compression of $\pi$ to $H$ yields the desired completely positive map $\Phi$ (Theorem 3.1). We further use this to construct a unital completely positive extension in the case where the $\Phi_{i}$ agree just as linear maps on $B$. Under the additional structure of [4], we can construct $\Phi$ so that (1.3) holds, and thus it coincides with Boca’s map. Actually we do more here. Given a sub-family of expectations $\{E_{j}:A_{j}\to B\}_{j\in J}$ for $J\subset I$, we can construct $\Phi$ that satisfies (1.3) for $i_{1},\dots,i_{n}\in J$ (Theorem 3.4).

2. Preliminaries

Let $I$ be an index set of arbitrary cardinality. Fix a family $\{A_{i}\}_{i\in I}$ of unital C*-algebras that contain a common unital C*-subalgebra $B$ in the sense that there are faithful unital imbeddings $\varepsilon_{i}:B\to A_{i}$ for $i\in I$. We denote by $\bigast_{B}A_{i}$ the $*$-algebraic amalgamated free product with canonical imbeddings $\varphi_{i}:A_{i}\to\bigast_{B}A_{i}$. The amalgamated free product $\check{\bigast}_{B}A_{i}$ of C*-algebras $\{A_{i}\}_{i\in I}$ is then its quotient completion with respect to the seminorm

[TABLE]

The supremum is finite since the $\pi\varphi_{i}$ are $*$-representations of $A_{i}$. The existence of $\check{\bigast}_{B}A_{i}$ is then routine; e.g. [7, p. 88]. Blackadar [3, Theorem 3.1] shows that such a $\pi$ can be constructed so that the $\pi\varphi_{i}$ are isometric. Therefore the canonical imbeddings $A_{i}\to\check{\bigast}_{B}A_{i}$ for $i\in I$ are isometric; and henceforth we will suppress their use. Notice that a $*$-representation $\pi$ of $\check{\bigast}_{B}A_{i}$ satisfies $\pi\varepsilon_{i}=\pi\varepsilon_{j}$ for all $i,j\in I$.

Pedersen [11] shows that if $B\subset C_{1}\subset A_{1}$ and $B\subset C_{2}\subset A_{2}$ are unital inclusions of C*-algebras, then the natural map of $C_{1}\check{\bigast}C_{2}$ into $A_{1}\check{\bigast}A_{2}$ given by the universal property is injective. An alternative proof was given later by Armstrong, Dykema, Exel and Li [1]. This result can be extended to free products of finitely many C*-algebras.

With Fuller, the authors gave a direct extension of these arguments in [7, Lemma 5.3.18]. Even though [7, Lemma 5.3.18] treats the finite case, the reasoning can be applied verbatim to tackle arbitrary families. We sketch the proof because we wish to establish some notation.

The following elementary fact will be used repeatedly. Suppose that $\Phi$ is a unital completely positive map of a C*-algebra $A$ into ${\mathcal{B}}(H)$, and $B$ is a C*-subalgebra of $A$ such that $\Phi|_{B}$ is a $*$-representation. Then given any Stinespring dilation $\pi:A\to{\mathcal{B}}(H\oplus H^{\prime})$ of $\Phi$, the space $H$ reduces $\pi(B)$. This follows because the compression of a $*$-representation to a subspace is multiplicative only when the subspace is invariant.

Lemma 2.1**.**

Let $B$ be a unital C-algebra. Let $\{A_{i}\}_{i\in I}$ be a family of unital C*-algebras with faithful unital imbeddings $\varepsilon_{i}:B\to A_{i}$, and suppose that there are $*$-representations $\rho_{i}:B\to{\mathcal{B}}(H_{i})$ for $i\in I$. Then there are a Hilbert space $K$ containing $\sum_{i\in I}^{\oplus}H_{i}$ and $*$-representations $\pi_{i}:A_{i}\to{\mathcal{B}}(K\ominus H_{i})$ such that the $\pi_{i}$ agree on $B$ in the sense that*

[TABLE]

Proof..

Recall that $S$ is the set of non-empty words in $I$ that contain no subwords of the form $ii$. For a word $i_{1}\cdots i_{n}$ in $I$, we set

[TABLE]

We will recursively define Hilbert spaces $H_{w}$ for $w\in S$ with $|w|\geq 2$, $*$-representations $\rho_{w}:B\to{\mathcal{B}}(H_{w})$, and for $u\in S$ and $i\neq s(u)$, we will construct a $*$-representation $\pi_{i,u}:A_{i}\to{\mathcal{B}}(H_{u}\oplus H_{ui})$ so that $\pi_{i,u}\varepsilon_{i}=\rho_{u}\oplus\rho_{ui}$.

By hypothesis, the $\rho_{j}$ are defined for $j\in I$. Consider them as unital completely positive maps of $B$ into ${\mathcal{B}}(H_{j})$. For each $i\neq j$, use Arveson’s extension theorem to extend $\rho_{j}$ to a completely positive map of $A_{i}$ into ${\mathcal{B}}(H_{j})$, and then apply Stinespring’s dilation theorem111 Alternatively we may use the more elementary result [10, Proposition 2.10.2]. For reasons that will become clear later, we do not follow this route. to obtain a $*$-representation $\pi_{i,j}:A_{i}\to{\mathcal{B}}(H_{j}\oplus H_{ji})$. By the remark preceding the lemma, $\pi_{i,j}\varepsilon_{i}(B)$ reduces $H_{j}$ (and hence also $H_{ji}$). So we may define a $*$-representation $\rho_{ji}:=P_{H_{ji}}\pi_{i,j}\varepsilon_{i}$. We then have $\pi_{i,j}\varepsilon_{i}=\rho_{j}\oplus\rho_{ji}$.

Now consider $k\geq 2$. Suppose that the $H_{w}$ are defined for $w\in S$ with $|w|\leq k$, that $\rho_{w}:B\to{\mathcal{B}}(H_{w})$ are $*$-representations, and that there are $*$-representations $\pi_{i,u}$ of $A_{i}$ on $H_{u}\oplus H_{ui}$ whenever $u\in S$ with $|u|<k$ and $s(u)\neq i$ such that $\pi_{i,u}\varepsilon_{i}=\rho_{u}\oplus\rho_{ui}$. If $|w|=k$ and $i\neq s(w)$, use Arveson’s extension theorem to extend $\rho_{w}$ to a completely positive map of $A_{i}$ into ${\mathcal{B}}(H_{w})$. Then apply Stinespring’s dilation theorem to obtain a $*$-representation $\pi_{i,w}:A_{i}\to{\mathcal{B}}(H_{w}\oplus H_{wi})$. By the remark preceding the lemma, $\pi_{i,w}\varepsilon_{i}(B)$ reduces $H_{w}$ (and hence also $H_{wi}$). So we may define a $*$-representation $\rho_{wi}:=P_{H_{wi}}\pi_{i,w}\varepsilon_{i}$. We then have $\pi_{i,w}\varepsilon_{i}=\rho_{w}\oplus\rho_{wi}$. This completes the induction.

Set $K=\sum_{w\in S}^{\oplus}H_{w}$. Define $\pi_{i}=\sum^{\oplus}_{s(w)\neq i}\pi_{i,w}$, which is a $*$- representation of $A_{i}$ on

[TABLE]

Moreover

[TABLE]

Thus $\rho_{i}\oplus\pi_{i}\varepsilon_{i}=\sum^{\oplus}_{u\in S}\rho_{u}$ is independent of $i$. ∎

As a consequence we get the imbedding of [11, 1, 7] for arbitrary families of C*-algebras.

Corollary 2.2**.**

Let $\{C_{i}\}_{i\in I}$ and $\{A_{i}\}_{i\in I}$ be families of unital C-algebras such that $B\subset C_{i}\subset A_{i}$ for a common unital C*-subalgebra $B$. Then $\check{\bigast}_{B}C_{i}\subset\check{\bigast}_{B}A_{i}$ via the natural inclusion map.*

In particular, if $J$ is a non-empty subset of $I$, then $\check{\bigast}_{B}^{i\in J}A_{i}\subset\check{\bigast}_{B}A_{i}$ via the natural inclusion map.

Proof..

For the first part, let $\sigma:\check{\bigast}_{B}C_{i}\to{\mathcal{B}}(H)$ be a faithful $*$-representation. We can find Hilbert spaces $H_{i}$ and extend every $\sigma_{i}:=\sigma|_{C_{i}}$ to a $*$-representation $\widetilde{\sigma}_{i}\colon A_{i}\to{\mathcal{B}}(H\oplus H_{i})$. Since $\sigma_{i}\varepsilon_{i}$ is a $*$-representation of $B$ on $H$, we can decompose $\widetilde{\sigma}_{i}\varepsilon_{i}=\sigma_{i}\varepsilon_{i}\oplus\rho_{i}$. Hence we can apply Lemma 2.1 to $\rho_{i}:B\to{\mathcal{B}}(H_{i})$ and obtain the $*$-representations $\pi_{i}\colon A_{i}\to{\mathcal{B}}(K\ominus H_{i})$ such that

[TABLE]

Define $\tau_{i}:A_{i}\to{\mathcal{B}}(H\oplus K)$ by $\tau_{i}=\widetilde{\sigma}_{i}\oplus\pi_{i}$. Since $\sigma_{i}\varepsilon_{i}=\sigma_{j}\varepsilon_{j}$, this construction yields

[TABLE]

The universal property of the amalgamated free product then yields a $*$-representation of $\check{\bigast}_{B}A_{i}$ that extends $\sigma$. On the other hand every $*$-representation of $\bigast_{B}A_{i}$ restricts to a $*$-representation of $\bigast_{B}C_{i}$. Hence the canonical inclusion map $\bigast_{B}C_{i}\hookrightarrow\bigast_{B}A_{i}$ extends to an isometry on their completions.

We can now apply this to the family $\{C_{i}\}_{i\in I}$ given by

[TABLE]

Notice that

[TABLE]

The second claim then follows, since the $*$-representations of $\bigast_{B}^{i\in J}A_{i}$ and $\bigast_{B}C_{i}$ coincide. ∎

The free product construction can be readily formulated when $A_{i}$ are possibly non-selfadjoint. They have been studied first by Duncan [9]. Their theory was later established in [7, Section 5.3], and exhibited in an alternative way by Dor-On and Salomon [8, Section 4]. Dor-On and Salomon [8, Proposition 4.3] develop similar dilation techniques and establish Corollary 2.2 in that generality.

3. Free products of ucp maps

We first establish that there is always a common extension of unital completely positive maps on each $A_{i}$ to the amalgamated free product when they restrict to a common $*$-representation on $B$. As this does not require expectations, it is a natural extension of Boca’s result.

Theorem 3.1**.**

Let $B$ be a unital C-algebra. Let $\{A_{i}\}_{i\in I}$ be a family of unital C*-algebras and let $\varepsilon_{i}:B\to A_{i}$ be faithful unital imbeddings. Let $\Phi_{i}:A_{i}\to{\mathcal{B}}(H)$ be unital completely positive maps which restrict to a common $*$-representation of $B$. Then there is a unital completely positive map $\Phi:\check{\bigast}_{B}A_{i}\to{\mathcal{B}}(H)$ such that $\Phi|_{A_{i}}=\Phi_{i}$ for all $i\in I$.*

Proof..

By hypothesis $\rho_{0}=\Phi_{i}\varepsilon_{i}$ is a $*$-representation of $B$ which is independent of $i\in I$. Use Stinespring’s Theorem to dilate each $\Phi_{i}$ to a $*$-representation $\sigma_{i}$ of $A_{i}$ on $H\oplus H_{i}$ such that $\Phi_{i}(a)=P_{H}\sigma_{i}(a)|_{H}$ for $a\in A_{i}$. Since $\Phi_{i}\varepsilon_{i}=\rho_{0}$ is a $*$-representation, it follows that $\sigma_{i}\varepsilon_{i}=\rho_{0}\oplus\rho_{i}$ is a direct sum of $*$-representations of $B$ on $H$ and $H_{i}$. Now we apply Lemma 2.1 to the family of $*$-representations $\rho_{i}:B\to{\mathcal{B}}(H_{i})$ for $i\in I$, and obtain $*$-representations $\pi_{i}:A_{i}\to{\mathcal{B}}(K\ominus H_{i})$ such that

[TABLE]

Then the $*$-representations

[TABLE]

agree on $B$, i.e. $\tau_{i}\varepsilon_{i}=\tau_{j}\varepsilon_{j}$. By the universal property of free products, there is a $*$-representation

[TABLE]

Then $\Phi=P_{H}\tau|_{H}$ is the required completely positive map. ∎

We can extend Theorem 3.1 to the case where the $\Phi_{i}$ agree just as linear maps on $B$.

Theorem 3.2**.**

Let $B$ be a unital C-algebra. Let $\{A_{i}\}_{i\in I}$ be a family of unital C*-algebras and let $\varepsilon_{i}:B\to A_{i}$ be faithful unital imbeddings. Let $\Phi_{i}:A_{i}\to{\mathcal{B}}(H)$ be unital completely positive maps which restrict to a common linear map of $B$. Then there is a unital completely positive map $\Phi:\check{\bigast}_{B}A_{i}\to{\mathcal{B}}(H)$ such that $\Phi|_{A_{i}}=\Phi_{i}$ for all $i\in I$.*

Proof..

By hypothesis $\Phi_{0}=\Phi_{i}\varepsilon_{i}$ is a unital completely positive map of $B$ into ${\mathcal{B}}(H)$ which is independent of $i\in I$. Let $\sigma_{i}:A_{i}\to{\mathcal{B}}(H\oplus H_{i})$ be a Stinespring dilation of each $\Phi_{i}$ and set $M_{i}=\overline{\sigma_{i}\varepsilon_{i}(B)H}$. This is the minimal reducing subspace for $\sigma_{i}\varepsilon_{i}(B)$, and thus determines a minimal Stinespring dilation $\rho_{i}$ of $\Phi_{0}$, namely

[TABLE]

Fix some $i_{0}\in I$. By uniqueness of the minimal dilation, there are unitary operators $U_{i}:M_{i_{0}}\to M_{i}$ such that

[TABLE]

Define unital completely positive maps of $A_{i}$ into ${\mathcal{B}}(M_{i_{0}})$ by

[TABLE]

Notice that they restrict to the common $*$-representation $\rho_{i_{0}}$ on $B$. Thus by Theorem 3.1, there is a unital completely positive map $\Psi:\check{\bigast}_{B}A_{i}\to{\mathcal{B}}(M_{i_{0}})$ such that $\Psi|_{A_{i}}=\Psi_{i}$. For $a\in A_{i}$ we have

[TABLE]

Therefore the compression $\Phi=P_{H}\Psi|_{H}$ is the required map. ∎

We can modify the construction of Theorem 3.1 to prove Boca’s result. We use the following lemma.

Lemma 3.3**.**

Begin with the same setup and notation as in Lemma 2.1 and its proof. Furthermore assume that there is a subset $J\subset I$ for which there are conditional expectations $E_{j}$ of $A_{j}$ onto $B$ for $j\in J$. Then there are $*$-representations $\pi_{i}:A_{i}\to{\mathcal{B}}(K\ominus H_{i})$ such that

[TABLE]

with the additional property that $\pi_{j}(a)H_{w}\subset H_{wj}$ whenever $a\in\ker E_{j}$ for $j\in J$ and $s(w)\neq j$.

Proof..

The only change to the proof of Lemma 2.1 is in the construction of the $*$-representations $\pi_{j,w}$ for $j\in J$ and $s(w)\neq j$. For these $(j,w)$ we will use the completely positive map $\rho_{w}E_{j}$ from $A_{j}$ into ${\mathcal{B}}(H_{w})$. Then use Stinespring’s Dilation Theorem to obtain the $*$-representation $\pi_{j,w}$ of $A_{j}$ into ${\mathcal{B}}(H_{w}\oplus H_{wj})$ such that $\rho_{w}E_{j}=P_{H_{w}}\pi_{j,w}|_{H_{w}}$. Since $\ker E_{j}$ is in the kernel of this completely positive map, we obtain the matrix form with respect to the decomposition $H_{w}\oplus H_{wj}$

[TABLE]

for $a\in\ker E_{j}$. This means that $\pi_{j}(a)H_{w}\subset H_{wj}$. ∎

We are now able to give the dilation theoretic proof that generalizes [4, Theorem 3.1]. This provides an explicit Stinespring dilation, which Boca accomplished in his setting in [5].

Theorem 3.4**.**

Let $\{A_{i}\}_{i\in I}$ be unital C-algebras containing a common unital C*-subalgebra $B$, and suppose that there are conditional expectations $E_{j}$ of $A_{j}$ onto $B$ for $j\in J\subset I$. Let $\Phi_{i}:A_{i}\to{\mathcal{B}}(H)$ be unital completely positive maps which restrict to a common $*$-representation of $B$. Then there is a unital completely positive map $\Phi:\check{\bigast}_{B}A_{i}\to{\mathcal{B}}(H)$ such that $\Phi|_{A_{i}}=\Phi_{i}$ for all $i\in I$, and*

[TABLE]

when all $j_{k}\in J$, $a_{k}\in\ker E_{j_{k}}\subset A_{j_{k}}$, and $j_{1}\neq\cdots\neq j_{n}$.

Proof..

The construction is identical to the proof of Theorem 3.1 except that we use the refinement in Lemma 3.3. Therefore we now have $*$-representations

[TABLE]

such that for $h\in H$ and $a\in A_{i}$,

[TABLE]

and in addition

[TABLE]

It remains only to verify the identity (3.1). By construction we have

[TABLE]

We will show by induction that for $h\in H$,

[TABLE]

when all $j_{k}\in J$, $a_{k}\in\ker E_{j_{k}}\subset A_{j_{k}}$, and $j_{1}\neq\cdots\neq j_{n}$. This holds for $n=1$. Assuming the result for $n-1$, we see that

[TABLE]

By induction, this holds for all $n$. Compression to $H$ yields that

[TABLE]

Acknowledgements**.**

The authors acknowledge support from the London Mathematical Society. Kenneth Davidson visited Newcastle University under a Scheme 2 LMS grant (Ref: 21602). The authors would like to thank Steven Power (Lancaster University) and Ivan Todorov (Queen’s University of Belfast) for supporting this UK visit.

We thank Chris Ramsey for asking a question which led to the inclusion of Theorem 3.2.

We also thank the referee for a meticulous reading of our paper.