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.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2021

Author dc:creator

• Amrutam, Tattwamasi

Advisor dc:contributor.advisor

• Kalantar, Mehrdad

Committee members dc:contributor.committeemember

• Blecher, David P.
• Ott, William
• Skalski, Adam G.

• Crossed products, C*-algebras

Statement dc:rights

• The 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).

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/10657/10237

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/10237