Uniform property $\Gamma$ for Crossed products by group actions with the Rokhlin-type properties

TL;DR

This paper demonstrates that for certain group actions with Rokhlin-type properties on a unital simple C*-algebra with uniform property Γ, both the crossed product and fixed point algebra also possess uniform property Γ.

Contribution

It establishes that uniform property Γ is preserved under crossed products and fixed point algebras for actions with Rokhlin-type properties.

Findings

Crossed products by finite group actions with weak Rokhlin property have uniform property Γ.

Fixed point algebras under finite group actions with weak Rokhlin property have uniform property Γ.

Crossed products by compact group actions with tracial Rokhlin property with comparison have uniform property Γ.

Abstract

In this paper, let $A$ be a unital separable simple infinite dimensional C*-algebra which has uniform property $\Gamma$. Let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a finite group which has the weak tracial Rokhlin property. Then we prove that the crossed product $A\rtimes_\alpha G$ and fixed point algebra $A^\alpha$ have uniform property $\Gamma$. Let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a second-countable compact group which has the tracial Rokhlin property with comparison. Then we prove that the crossed product $A\rtimes_\alpha G$ and fixed point algebra $A^\alpha$ have uniform property $\Gamma$.