On the structure of the Cuntz semigroup in (possibly) nonunital C*-algebras

TL;DR

This paper investigates the structure of the Cuntz semigroup in both unital and nonunital simple C*-algebras, revealing conditions under which it can be reconstructed from other invariants and providing examples in subhomogeneous algebras.

Contribution

It establishes criteria for recovering the Cuntz semigroup from the Murray-von Neumann semigroup and traces in nonunital algebras, extending understanding of their structure.

Findings

Unital simple C*-algebras with finite extreme tracial boundary have positive operators of all ranks.

Criteria are provided for the functorial recovery of the Cuntz semigroup in nonunital algebras.

Approximately subhomogeneous algebras of slow dimension growth satisfy these criteria.

Abstract

We examine the ranks of operators in semi-finite C*-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple C*-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth.