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

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 $\Gamma$.