Kalantar, Mehrdad2022-06-302022-06-30May 20212021-05May 2021Portions of this document appear in: Amrutam, Tattwamasi, and Mehrdad Kalantar. "On simplicity of intermediate-algebras." Ergodic Theory and Dynamical Systems 40, no. 12 (2020): 3181-3187; and in: Amrutam, Tattwamasi. "On Intermediate Subalgebras of C*-simple Group Actions." International Mathematics Research Notices 2021, no. 21 (2021): 16193-16204; and in: Amrutam, Tattwamasi, and Dan Ursu. "A generalized Powers averaging property for commutative crossed products." Transactions of the American Mathematical Society 375, no. 03 (2022): 2237-2254.https://hdl.handle.net/10657/10237This 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.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).Crossed products, C*-algebras2022-06-30Thesisborn digital