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.


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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