Centrally pure C*-algebras

TL;DR

This paper characterizes $\\mathcal{Z}$-stability of separable C*-algebras through the purity of their central sequence algebras, providing new equivalences and generalizations in the structure theory of C*-algebras.

Contribution

It establishes that $\\mathcal{Z}$-stability is equivalent to the purity of various central sequence algebras, extending previous results to a more general setting.

Findings

$\\mathcal{Z}$-stability iff uncorrected central sequence algebra is pure

$\\mathcal{Z}$-stability iff Kirchberg's central sequence algebra is pure

Generalization to relative central sequence algebras for all separable subalgebras

Abstract

We show that a separable C*-algebra $A$ is $\mathcal{Z}$-stable if and only if its uncorrected central sequence algebra $A' \cap A_{\mathcal{U}}$ is pure, if and only if Kirchberg's central sequence algebra $F(A)$ is pure. More generally, we show that a C*-algebra $A$ is separably $\mathcal{Z}$-stable if and only if the relative central sequence algebra $B' \cap A_{\mathcal{U}}$ is pure for every separable subalgebra $B \subseteq A_{\mathcal{U}}$.