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I am currently writing a project wherein I need two identifications. Let's fix some notation. Let $\Gamma$ be a discrete group acting on a unital C*-algebra A by $\ast$-automorphisms $\alpha: A \rightarrow \text{Aut}(A)$ and let per usual $C_c(\Gamma,A)$ be the involutive algebra generated by unitaries $\lbrace u_s \rbrace_{s\in \Gamma}$ and $A$ such that we have $\alpha_s(a) = u_s a u_s^*$ for all $s\in \Gamma$. The reduced crossed product of the triple $(A,\alpha, \Gamma)$ is defined to be the norm-closure of $C_c(\Gamma,A)$ under the image of the faithful representation \begin{equation} \pi \times \lambda: C_c(\Gamma,A) \rightarrow B(H)\otimes \ell^2(\Gamma), \end{equation}

where $\pi: A \rightarrow B(H)$ is any faithful representation and $\lambda$ is the left-regular representation of $\Gamma$. My questions concern two identifications.\

(1) Let $H$ be another group acting on $G$, via $\varphi: H\rightarrow \text{Aut}(\Gamma)$ and form the semidirect product $G\rtimes H$. If $G\times H$ acts on $A$, I want the identification $(A\rtimes \Gamma) \rtimes H\cong A\rtimes (\Gamma \rtimes H)$.

(2) I am looking for an identification along the lines of the $$ A\otimes A \rtimes_{\alpha \otimes \alpha} \Gamma^2 \cong A\rtimes_\alpha \Gamma \otimes A\rtimes_\alpha \Gamma. $$

Here $\alpha \otimes \alpha$ denotes the action of $\Gamma \times \Gamma$ on $A \otimes A$ given by $(\alpha_s \otimes \alpha_t)(a\otimes b)=\alpha_s(a) \otimes \alpha_t(b)$ for all $s,t\in \Gamma$and $a\otimes b\in A\otimes A$.

I believe (2) should follow from more or less direct computations when representing two generic elements via the representations $(\pi \times \lambda) \otimes (\pi \times \lambda)$ and $(\pi \otimes \pi) \times \lambda'$ where $\lambda'$ is the left-regular representation on $\Gamma \times \Gamma$. I would appreciate any comment or confirmation to this ( I can write the computations down if needed).

For number (1), I am slightly confused. I suspect one would have to do something similar as in (2), but one needs to describe how $H$ and $\Gamma \rtimes H$ will act on $A\rtimes \Gamma$. My initial thought was to consider the action \begin{equation} \bar{\varphi}: H\rightarrow \text{Aut}(C_c(\Gamma,A)); \ \bar{\varphi}_h(au_s) = au_{\varphi_h(s)} \end{equation}

and extend to $A\rtimes \Gamma$ by continuity, while letting $G\rtimes H$ act on $C_c(\Gamma, A)$by the action \begin{equation} \bar{\beta}_{(g,h)}(au_s) = \alpha_g(a)u_{\varphi_h(s)} \end{equation}

Am I even on the right track here? Any hints or comments will be appreciated greatly. Thanks in advance.

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Question (1) seems to be missing from your post.

For the identification (2), the unitary $U : \ell^2(\Gamma \times \Gamma) \to \ell^2(\Gamma) \otimes \ell^2(\Gamma)$ defined by $\delta_{(g,h)} \mapsto \delta_g \otimes \delta_h$ intertwines $\lambda'$ and $\lambda \otimes \lambda$. Let $F : H \otimes \ell^2(\Gamma) \to \ell^2(\Gamma) \otimes H$ be the flip unitary. I think you should be able to check by direct computation that $(1_H \otimes F \otimes 1_{\ell^2(\Gamma)}) \circ (1_{H \otimes H} \otimes U)$ intertwines $(\pi \otimes \pi) \times \lambda'$ and $(\pi \times \lambda) \otimes (\pi \times \lambda)$.

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  • $\begingroup$ Oh yes I am terribly sorry, I mistakely erased it. It should appear now. But I believe you find (2) to be a correct statement? And thanks. $\endgroup$ – Munk Apr 2 '16 at 14:41
  • $\begingroup$ Yes, I believe that (2) is a correct statement, but I have not checked the details. I think you're on the right track with (1); those look like reasonable formulas. I'd guess you can do the same sort of thing discussed for (2), using the canonical unitary isomorphism $\ell^2(\Gamma \times H) \cong \ell^2(\Gamma) \otimes \ell^2(H)$ to write down a unitary that intertwines your two regular representations. $\endgroup$ – Aidan Sims Apr 2 '16 at 14:51

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