Ranks of soft operators in nowhere scattered C*-algebras

TL;DR

This paper investigates the structure of soft operators in C*-algebras with the Global Glimm Property, revealing how their ranks relate to the Cuntz semigroup and implications for the algebra's comparison radius.

Contribution

It demonstrates that in such C*-algebras, the rank of any operator can be realized by a soft operator, and establishes a retraction onto the soft part of the Cuntz semigroup.

Findings

Ranks of all operators can be realized by soft operators.

The radius of comparison is determined by the soft part of the Cuntz semigroup.

A retraction exists from the Cuntz semigroup onto its soft part.

Abstract

We show that for C*-algebras with the Global Glimm Property, the rank of every operator can be realized as the rank of a soft operator, that is, an element whose hereditary sub-C*-algebra has no nonzero, unital quotients. This implies that the radius of comparison of such a C*-algebra is determined by the soft part of its Cuntz semigroup. Under a mild additional assumption, we show that every Cuntz class dominates a (unique) largest soft Cuntz class. This defines a retract from the Cuntz semigroup onto its soft part, and it follows that the covering dimensions of these semigroups differ by at most $1$.