Structure of Intermediate C*-subalgebras of discrete group actions

This thesis deals with the structure of intermediate $C\ast$-sub-algebras $B$, either of the form $C\lambda \ast (\Gamma )\subseteq B\subseteq A⋊r\Gamma$ or of the type $C(Y)⋊r\Gamma \subseteq B\subseteq C(X)⋊r\Gamma$. We begin by investigating the ideal structure of intermediate $C\ast$-sub-algebras $B$ of the form $C\lambda \ast (\Gamma )\subseteq B\subseteq A⋊r\Gamma$ for commutative unital $\Gamma$-simple $\Gamma$-$C\ast$-algebras $A$. In particular, we show that if $\Gamma$ is a $C\ast$-simple group, then every such intermediate $C\ast$-sub-algebra $B$ is simple. Continuing our perusal, we find examples of inclusions $C\lambda \ast (\Gamma )\subseteq A⋊r\Gamma$ for which every intermediate $C\ast$-sub-algebra $B$ of the form $C\lambda \ast (\Gamma )\subseteq B\subseteq A⋊r\Gamma$ is a crossed product. We show that for a large class of actions $\Gamma \curvearrowright A$ of $C\ast$-simple groups $\Gamma$ on unital $C\ast$-algebras $A$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C\ast$-sub-algebra $B$, $C\lambda \ast (\Gamma )\subseteq B\subseteq A⋊r\Gamma$, is a crossed product. On the von Neumann algebraic side, we show that for every non-faithful action of a acylindrically hyperbolic $C\ast$-simple group $\Gamma$ on a von Neumann algebra $M$ with separable predual, every intermediate vNa $N$, $L(\Gamma )\subseteq N\subseteq M⋊\Gamma$ is a crossed product vNa. Finally, we inquire into the ideal structure of intermediate $C\ast$-sub-algebras $B$ of the form $C(Y)⋊r\Gamma \subseteq B\subseteq C(X)⋊r\Gamma$ for an inclusion of unital $\Gamma$-simple $\Gamma$-$C\ast$-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 $C(X)⋊r\Gamma$. As an application, we show that every intermediate $C\ast$-sub-algebras $B$, $C(Y)⋊r\Gamma \subseteq B\subseteq C(X)⋊r\Gamma$ is simple whenever $C(Y)⋊r\Gamma$ is simple.