  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.
