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.
Ramon Antoine. Francesc Perera. Leonel Robert. Hannes Thiel. "C*-algebras of stable rank one and their Cuntz semigroups." Duke Math. J. 171 (1) 33 - 99, 15 January 2022. https://doi.org/10.1215/00127094-2021-0009