An exercise about reduced crossed product of a C*-dynamical system

Question

Here is an Exercise in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa.

Exercise 4.1.3. Let $A$ and $B$ be two C*-algebras and $\Gamma$ be a discrete group. If $\alpha:\Gamma\rightarrow \mathrm{Aut}(A)$ is an action and $\tau\otimes\alpha:\Gamma\rightarrow Aut(B\otimes A)$ is defined by $(\tau\otimes\alpha)_{g}=\mathrm{id}_{B}\otimes\alpha_{g}$, then $$(B\otimes A)\rtimes_{\tau\otimes\alpha, r}\Gamma\cong B\otimes(A\rtimes_{\alpha, r}\Gamma).$$

Here, the "$B\otimes A$" denote the tensor product equiped with minimal norm. And $A\rtimes_{\alpha, r}\Gamma$ denotes the reduced crossed product of a C*-dynamical system $(A, \Gamma, \alpha)$, which is the norm closure of the image of a regular representation $C_{c}(\Gamma, A)\rightarrow B(H\otimes l^{2}(\Gamma))$. While the $C_{c}(\Gamma, A)$ denotes the linear space of finitely supported functions on $\Gamma$ with values in $A$.

Answer 1

This looks like an "associativity" property. Both algebras live in $B(H_B\otimes H_A\otimes \ell^2(\Gamma))$, so the topology is the same.

At the pre-closure level, you should convince yourself that $C_c(\Gamma,B\odot A)$ and $B\odot C_c(\Gamma,A)$ are equal, and that their closures give the two algebras you want to consider.

Answer 2

I know this is an old question, but I wanted to share my solution for this problem. Here goes.

---

Recall that if $(A, \Gamma, \alpha)$ is a $C^*$-dynamical system, the reduced cross product $A \rtimes_{\alpha,r}\Gamma$ is defined as follows:

Consider a faithful representation $\kappa_A: A \hookrightarrow B(H)$. We define a new representation $$\pi_{\kappa_A}: A \to B(H\otimes \ell^2(\Gamma))$$ by $\pi_{\kappa_A}(a)(\xi \otimes \delta_g) = \kappa_A(\alpha_{g^{-1}}(a))\xi \otimes \delta_g$. Next, we define $$\Psi_{\kappa_A}: C_c(\Gamma,A) \to B(H \otimes \ell^2(\Gamma)): \sum_{s\in \Gamma} a_s s \mapsto \sum_{s \in \Gamma} \pi_{\kappa_A}(a_s)(1 \otimes \lambda_s)$$ where $\lambda: \Gamma \to B(H)$ is the left regular representation. This is a $*$-representation and thus we obtain a $C^*$-norm $$\|x\|_r:= \|\psi_{\kappa_A}(x)\|$$ on $C_r(\Gamma,A)$, which does not depend on the choice of faithful representation $\kappa_A: A \hookrightarrow B(H)$. If we complete $C_c(\Gamma,A)$ with respect to this norm, we obtain a new $C^*$-algebra, called the reduced cross product of $A$ and $\Gamma$, and denoted by $A \rtimes_{\alpha,r} \Gamma$.

Note that by definition, $\Psi_{\kappa_A}: C_c(\Gamma,A) \to B(H\otimes \ell^2(\Gamma))$ becomes an isometry with respect to this norm, so we have a canonical (when a faithful representation of $A$ is fixed) faithful representation $$\Psi_{\kappa_A}: A \rtimes_{\alpha,r}\Gamma \hookrightarrow B(H\otimes \ell^2(\Gamma)).$$

---

Next, note that we have a canonical $*$-algebra morphism $$B \odot C_c(\Gamma, A) \to C_c(\Gamma, B \otimes A) \subseteq (B \otimes A)\rtimes_{\tau \otimes \alpha,r} \Gamma : b \otimes \sum_{s}a_s s \mapsto \sum_s (b\otimes a_s) s.$$

We show that this extends to an isometric $*$-morphism $$B\otimes (A \rtimes_{\alpha,r} \Gamma) \to (B\otimes A)\rtimes_{\tau \otimes \alpha,r}\Gamma.$$

For this, consider faithful representations $\kappa_A: A \hookrightarrow B(H_A)$ and $\kappa_B: B \hookrightarrow B(H_B)$.

Then we have \begin{align*}\|\sum_i b_i \otimes \left(\sum_{s \in \Gamma} a_s^i s\right)\|_{min}&= \|\sum_{i,s} b_i \otimes a_s^i s\|_{min}\\ &=\|\sum_{i,s} \kappa_{B}(b_i) \otimes \Psi_{\kappa_A}(a_s^i s)\|\\ &= \|\sum_{i,s}\kappa_B(b_i) \otimes \pi_{\kappa_A}(a_s^i)(1 \otimes \lambda_s)\|\end{align*}

On the other hand, we have a representation $\kappa_B \otimes \kappa_A: B \otimes A \hookrightarrow B(H_B \otimes H_A)$, so \begin{align*}\|\sum_i \sum_s (b_i \otimes a_s^i)s\|_r &= \|\sum_s (\sum_i b_i \otimes a_s^i)s\|\\ &= \|\sum_s \sum_i \pi_{\kappa_B\otimes \kappa_A}(b_i \otimes a_s^i) (1 \otimes \lambda_s) \|\end{align*}

We then observe that $\pi_{\kappa_B\otimes \kappa_A}(b_i \otimes a_s^i) (1 \otimes \lambda_s) = \kappa_B(b_i) \otimes \pi_{\kappa_A}(a_s^i)(1 \otimes \lambda_s)$ as operators in $B(H_B\otimes H_A \otimes \ell^2(\Gamma))$, from which it follows that the aforementioned map is an isometry, and we obtain the desired extension.

Finally, note that the image of the extension $B \otimes (A\rtimes_{\alpha,r} \Gamma) \to (B \otimes A) \rtimes_{\tau \otimes \alpha,r} \Gamma$ contains $C_c(\Gamma,B \odot A)$. Since this set is dense in the codomain and the range of a surjection is closed, we deduce that our map is a $*$-isomorphism and the proof is finished.