Crossed products of Hilbert pro-C∗-bimodules and associated pro-C∗-algebras

Category: Vol 32/2016 no. 2 Tags: pro-$C^{\ast }$-algebras, Hilbert pro-$C^{\ast }$-bimodules, crossed products

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An action $\left( \gamma ,\alpha \right)$ of a locally compact group $G$ on a Hilbert pro- $C^{\ast }$ -bimodule $\left( X,A\right)$ induces an action $\gamma \times \alpha$ of $G$ on $A\times _{X}\mathbb{Z}$ the crossed product of $A$ by $X$ . We show that if $\left( \gamma ,\alpha \right)$ is an inverse limit action, then the crossed product of $A\times _{\alpha }G$ by $X\times_{\gamma }G$ respectively of $A\times _{\alpha ,r}G$ by $X\times_{\gamma ,r}G$ is isomorphic to the full crossed product of $A\times _{X}\mathbb{Z}$ by $\gamma \times \alpha$ respectively the reduced crossed product of $A\times _{X}\mathbb{Z}$ by $\gamma \times \alpha$ .