  We show that for C*-algebras with the Global Glimm Property, the rank of
every operator can be realized as the rank of a soft operator, that is, an
element whose hereditary sub-C*-algebra has no nonzero, unital quotients. This
implies that the radius of comparison of such a C*-algebra is determined by the
soft part of its Cuntz semigroup.
  Under a mild additional assumption, we show that every Cuntz class dominates
a (unique) largest soft Cuntz class. This defines a retract from the Cuntz
semigroup onto its soft part, and it follows that the covering dimensions of
these semigroups differ by at most $1$.

Operator Algebras