Tracial oscillation zero and Z-stability

TL;DR

This paper establishes that certain simple amenable C*-algebras are Z-stable if and only if they have strict comparison for positive elements, extending understanding beyond finite-dimensional tracial bases.

Contribution

It proves the equivalence of Z-stability and strict comparison for a broad class of non-unital, simple amenable C*-algebras with complex tracial boundaries.

Findings

Z-stability characterized by strict comparison

Applicable to C*-algebras with non-finite-dimensional tracial bases

Extends classification results to more general tracial boundary structures

Abstract

Let $A$ be a (not necessarily unital) separable non-elementary simple amenable C*-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies certain condition (C). We show that $A$ is ${\cal Z}$-stable if and only if $A$ has strict comparison for positive elements. Extremal boundaries of simplexes which satisfy condition (C) may contain countable disjoint unions of $n$-dimensional cubes ($n\in \N$) as a subset.