RANKS OF SOFT OPERATORS IN NOWHERE SCATTERED C^∗-ALGEBRAS

Journal article, 2024

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.