Centrally pure C*-algebras

TL;DR

This work establishes a sharp regularity criterion for $\\mathcal{Z}$-stability of a separable $C^*$-algebra by translating it into pureness of central sequence algebras. It proves that $A$ is $\\mathcal{Z}$-stable if and only if the central sequence algebra $A' \\cap A_{\\\mathcal{U}}$ is pure, equivalently if Kirchberg's central sequence algebra $F(A)$ is pure, and it extends the criterion to relative central sequence algebras for separable subalgebras of $A_{\\\mathcal{U}}$. The paper also develops absorption theory for strongly self-absorbing algebras, providing several equivalent characterizations of $D$-stability and showing how divisibility properties in central sequence algebras control, and in fact detect, $\\\mathcal{Z}$-stability. Additionally, it introduces central divisibility notions and demonstrates that almost divisibility implies $\\\\mathcal{Z}$-stability, while noting open questions about weaker forms of divisibility such as functional divisibility.

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 $B \subseteq A_{\mathcal{U}}$.

Theorems & Definitions (32)

• Theorem 1: see \ref{['prp:CharZStable']}
• Theorem 2: see \ref{['prp:CharSepZStable']}
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof
• Definition 2.5