On topologically zero-dimensional morphisms

TL;DR

This paper characterizes zero-dimensional $^*$-homomorphisms with nuclear dimension zero, showing they can be approximated via AF-algebras and providing invariants and conditions for such maps, especially in real rank zero contexts.

Contribution

It introduces a characterization of zero-dimensional $^*$-homomorphisms as those approximately factorized through AF-algebras and analyzes their invariants and embeddings.

Findings

Zero-dimensional $^*$-homomorphisms can be approximated through AF-algebras.

Obstructions for the total invariant of zero-dimensional morphisms are identified.

Nuclear dimension zero is determined by the total invariant in real rank zero cases.

Abstract

We investigate $^*$-homomorphisms with nuclear dimension equal to zero. In the framework of classification of $^*$-homo-morphisms, we characterise such maps as those that can be approximately factorised through an AF-algebra. Along the way, we obtain various obstructions for the total invariant of zero-dimensional morphisms and show that in the presence of real rank zero, nuclear dimension zero can be completely determined at the level of the total invariant. We end by characterising when unital embeddings of $\mathcal{Z}$ have nuclear dimension equal to zero.