Hereditary uniform property $\Gamma$

TL;DR

This paper investigates the uniform property $mma$ in separable simple $C^*$-algebras, establishing its implications for tracial oscillation, stable rank, hereditary subalgebras, and ${al}$-stability.

Contribution

It introduces hereditary uniform property $mma$ and links it to important structural properties like ${al}$-stability in non-exact, simple, amenable $C^*$-algebras.

Findings

Algebras with uniform property $mma$ have tracial approximate oscillation zero.

Such algebras have stable rank one.

Hereditary subalgebras also possess a version of uniform property $mma$.

Abstract

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.