Extensions of pure C*-algebras

TL;DR

This paper investigates the conditions under which pure C*-algebras maintain their purity when extended or decomposed, and applies these results to stable multiplier algebras of free group C*-algebras.

Contribution

It establishes a characterization of pure C*-algebras via their ideals and quotients, and explores the preservation of comparison and divisibility properties in extensions.

Findings

A C*-algebra is pure if and only if its ideal and quotient are pure.

Permanence of comparison and divisibility properties in extensions is studied.

Stable multiplier algebras of reduced free group C*-algebras are shown to be pure.

Abstract

Given a closed ideal $I$ in a C*-algebra $A$, we show that $A$ is pure if and only if $I$ and $A/I$ are pure. More generally, we study permanence of comparison and divisibility properties when passing to extensions. As an application we show that stable multiplier algebras of reduced free group C*-algebras are pure.