Hereditary uniform property Γ

We study the uniform property Γ for separable simple C*-algebras which have quasitraces and may not be exact. We show that a stably finite separable simple C*-algebra A with the strict comparison and uniform property Γ has tracial approximate oscillation zero and stable rank one. Moreover in this case, its hereditary C*-subalgebras also have a version of uniform property Γ. If a separable non-elementary simple amenable C*-algebra A with strict comparison has this hereditary uniform property Γ, then A is Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{Z}$$\end{document}-stable.