$\mathcal Z$-stability and finite dimensional tracial boundaries

TL;DR

This paper establishes that certain nuclear $C^*$-algebras with finite-dimensional tracial boundaries are $ ext{Z}$-stable, using properties of order zero maps and strict comparison, advancing classification theory.

Contribution

It proves that strict comparison implies $ ext{Z}$-stability for simple nuclear $C^*$-algebras with finite-dimensional tracial boundaries, under specific conditions.

Findings

Existence of uniformly tracially large order zero maps

Strict comparison implies $ ext{Z}$-stability in this setting

Extension of classification results to broader class of $C^*$-algebras

Abstract

We show that a simple separable unital nuclear nonelementary $C^*$-algebra whose tracial state space has a compact extreme boundary with finite covering dimension admits uniformly tracially large order zero maps from matrix algebras into its central sequence algebra. As a consequence, strict comparison implies $\Z$-stability for these algebras.