Structure of Intermediate C*-subalgebras of discrete group actions
This thesis deals with the structure of intermediate $C^*$-sub-algebras $\mathcal{B}$, either of the form $C_{\lambda}^*(\Gamma)\subseteq\mathcal{B}\subseteq\mathcal{A}\rtimes_r\Gamma$ or of the type $C(Y)\rtimes_r\Gamma\subseteq\mathcal{B}\subseteq C(X)\rtimes_r\Gamma$. We begin by investigating the ideal structure of intermediate $C^*$-sub-algebras $\mathcal{B}$ of the form $C_{\lambda}^*(\Gamma)\subseteq\mathcal{B}\subseteq\mathcal{A}\rtimes_r\Gamma$ for commutative unital $\Gamma$-simple $\Gamma$-$C^*$-algebras $\mathcal{A}$. In particular, we show that if $\Gamma$ is a $C^*$-simple group, then every such intermediate $C^*$-sub-algebra $\mathcal{B}$ is simple. Continuing our perusal, we find examples of inclusions $C_{\lambda}^*(\Gamma)\subseteq \mathcal{A}\rtimes_r\Gamma$ for which every intermediate $C^*$-sub-algebra $\mathcal{B}$ of the form $C_{\lambda}^*(\Gamma)\subseteq\mathcal{B}\subseteq\mathcal{A}\rtimes_r\Gamma$ is a crossed product. We show that for a large class of actions $\Gamma\curvearrowright\mathcal{A}$ of $C^*$-simple groups $\Gamma$ on unital $C^*$-algebras $\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C^*$-sub-algebra $\mathcal{B}$, $C_{\lambda}^*(\Gamma)\subseteq\mathcal{B}\subseteq\mathcal{A}\rtimes_r\Gamma$, is a crossed product. On the von Neumann algebraic side, we show that for every non-faithful action of a acylindrically hyperbolic $C^*$-simple group $\Gamma$ on a von Neumann algebra $\mathcal{M}$ with separable predual, every intermediate vNa $\mathcal{N}$, $L(\Gamma)\subseteq\mathcal{N}\subseteq\mathcal{M}\rtimes\Gamma$ is a crossed product vNa. Finally, we inquire into the ideal structure of intermediate $C^*$-sub-algebras $\mathcal{B}$ of the form $C(Y)\rtimes_r\Gamma\subseteq\mathcal{B}\subseteq C(X)\rtimes_r\Gamma$ for an inclusion of unital $\Gamma$-simple $\Gamma$-$C^*$-algebras $C(Y)\subset C(X)$. We introduce a notion of generalized Powers' averaging and show that it is equivalent to the simplicity of the crossed product $C(X)\rtimes_r\Gamma$. As an application, we show that every intermediate $C^*$-sub-algebras $\mathcal{B}$, $C(Y)\rtimes_r\Gamma\subseteq\mathcal{B}\subseteq C(X)\rtimes_r\Gamma$ is simple whenever $C(Y)\rtimes_r\Gamma$ is simple.
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