Pure *-homomorphisms

TL;DR

This paper introduces the concept of pureness for *-homomorphisms and cpc order-zero maps, exploring their properties, examples, and factorization behavior within C*-algebras, with implications for understanding their structure.

Contribution

It defines and studies pureness for *-homomorphisms and cpc order-zero maps, providing examples and establishing factorization results through pure C*-algebras.

Findings

Pure maps can be factored through pure C*-algebras.

Composition of pure maps remains pure up to Cuntz equivalence.

Several factorization results at the C*-algebra level are obtained.

Abstract

We introduce and study a notion of pureness for *-homomorphisms and, more generally, for cpc. order-zero maps. After providing several examples of pureness, such as "$\mathcal{Z}$-stable"-like maps, we focus on the question of when pure maps factor through a pure C*-algebra. We show that, up to Cuntz equivalence, any composition of two pure maps factors through a pure object. This is used to obtain several factorization results at the level of C*-algebras.