Duke Math. J. 171 (1), 33-99, (15 January 2022) DOI: 10.1215/00127094-2021-0009
Ramon Antoine, Francesc Perera, Leonel Robert, Hannes Thiel
KEYWORDS: Cuntz semigroup, C*-algebra, stable rank one, Hilbert C*-module, semilattice, 46L05, 06B35, 06F05, 19K14, 46L08, 46L35
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.