Continuous functions over a pure C*-algebra

TL;DR

This paper investigates conditions under which the algebra of continuous functions over a compact metric space with values in a pure C*-algebra remains pure, and explores permanence properties of pureness under tensor products.

Contribution

It establishes that $C(X,A)$ is pure when $A$ is simple or has stably finite quotients, and shows tensor products with ASH-algebras preserve pureness.

Findings

$C(X,A)$ is pure if $A$ is simple or has stably finite quotients.

Tensor products of such $A$ with ASH-algebras are also pure.

Provides permanence results for the property of pureness in C*-algebras.

Abstract

Let $X$ be a compact metric space, and let $A$ be a pure $\mathrm{C}^*$-algebra. We show that $C(X,A)$ is pure whenever $A$ is simple; or every quotient of $A$ is stably finite (e.g., $A$ has stable rank one). Using permanence properties of pureness, we prove that the tensor product of any such $A$ with any ASH-algebra is pure.