  We study the uniform property $\Gamma$ 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 strict
comparison and uniform property $\Gamma$ has tracial approximate oscillation
zero and stable rank one. Moreover in this case, its hereditary
$C^*$-subalgebras also have a version of uniform property $\Gamma.$ If a
separable non-elementary simple amenable $C^*$-algebra $A$ with strict
comparison has this hereditary uniform property $\Gamma,$ then $A$ is ${\cal
Z}$-stable.
