C*-algebras of stable rank one and their Cuntz semigroups

TL;DR

This paper explores the structure of C*-algebras with stable rank one through their Cuntz semigroups, providing solutions to longstanding conjectures and problems in the field.

Contribution

It introduces new structural insights into the Cuntz semigroup of stable rank one C*-algebras, solving key conjectures and problems in the area.

Findings

Confirmed the Blackadar-Handelman conjecture for stable rank one C*-algebras.

Resolved the Global Glimm Halving problem for this class.

Established realization of functions on 2-quasitraces as ranks of Cuntz semigroup elements.

Abstract

The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: We answer affirmatively, for the class of stable rank one C*-algebras, a conjecture by Blackadar and Handelman on dimension functions, the Global Glimm Halving problem, and the problem of realizing functions on the cone of 2-quasitraces as ranks of Cuntz semigroup elements. We also gain new insights into the comparability properties of positive elements in C*-algebras of stable rank one.