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

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