Hereditary uniform property $\Gamma$

Abstract

We study the uniform property $\Gamma$ for separable simple $C^{\ast }$-algebras which have quasitraces and may not be exact. We show that a stably finite separable simple $C^{\ast }$-algebra $A$ with strict comparison and uniform property $\Gamma$ has tracial approximate oscillation zero and stable rank one. Moreover in this case, its hereditary $C^{\ast }$-subalgebras also have a version of uniform property $\Gamma .$ If a separable non-elementary simple amenable $C^{\ast }$-algebra $A$ with strict comparison has this hereditary uniform property $\Gamma ,$ then $A$ is $\mathcal{Z}$-stable.