Completely positive maps of order zero

TL;DR

This paper characterizes order zero completely positive maps between C*-algebras, establishing their structure, properties, and relationships with *-homomorphisms and Cuntz semigroups, advancing understanding in operator algebra theory.

Contribution

It provides a structure theorem for order zero maps and links them to *-homomorphisms from the cone over the domain, revealing their fundamental properties.

Findings

Order zero maps correspond to *-homomorphisms from the cone over the domain.

Tensor products of order zero maps are again order zero.

Order zero maps induce ordered semigroup morphisms between Cuntz semigroups.

Abstract

We say a completely positive contractive map between two C*-algebras has order zero, if it sends orthogonal elements to orthogonal elements. We prove a structure theorem for such maps. As a consequence, order zero maps are in one-to-one correspondence with *-homomorphisms from the cone over the domain into the target algebra. Moreover, we conclude that tensor products of order zero maps are again order zero, that the composition of an order zero map with a tracial functional is again a tracial functional, and that order zero maps respect the Cuntz relation, hence induce ordered semigroup morphisms between Cuntz semigroups.