A Galois correspondence for reduced crossed products of unital simple C$^*$-algebras by discrete groups

TL;DR

This paper establishes a Galois correspondence between subgroups of a discrete group acting by outer automorphisms on a unital simple C*-algebra and intermediate C*-algebras within the reduced crossed product, extending to twisted systems.

Contribution

It introduces a Galois correspondence for reduced crossed products of unital simple C*-algebras by discrete groups, including twisted cases, which was previously unestablished.

Findings

Galois correspondence between subgroups and intermediate C*-algebras

Extension of results to twisted dynamical systems

Framework for analyzing automorphism group actions on C*-algebras

Abstract

Let a discrete group $G$ act on a unital simple C$^*$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes_{\alpha,r}H$ between subgroups of $G$ and C$^*$-algebras $B$ satisfying $A\subseteq B \subseteq A\rtimes_{\alpha,r}G$, where $A\rtimes_{\alpha,r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\alpha,\sigma)$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes^\sigma_{\alpha,r}G$.