Quasidiagonal traces and crossed products

TL;DR

This paper investigates how quasidiagonality of traces on a $C^*$-algebra is preserved under crossed products by finite group actions with the weak tracial Rokhlin property, and explores related structural properties.

Contribution

It proves that quasidiagonality of traces is preserved under crossed products with certain finite group actions and examines the stability of this property under Rokhlin dimension.

Findings

All traces on the crossed product are quasidiagonal if all traces on the original algebra are.

Finite decomposition rank behavior under crossed products is characterized.

Quasidiagonality of traces is stable under crossed products with finite Rokhlin dimension.

Abstract

Let $A$ be a simple, exact, separable, unital $C^*$-algebra and let $\alpha \colon G \rightarrow Aut(A)$ be an action of a finite group $G$ with the weak tracial Rokhlin property. We show that every trace on $A \rtimes_{\alpha} G$ is quasidiagonal provided that all traces on $A$ are quasidiagonal. As an application, we study the behavior of finite decomposition rank under taking crossed products by finite group actions with the weak tracial Rokhlin property. Moreover, we discuss the stability of the property that all traces are quasidiagonal under taking crossed products of finite group actions with finite Rokhlin dimension with commuting towers.