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$\mathrm {C}^*$-ALGEBRAS
We show that for 
$\mathrm {C}^*$-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-
$\mathrm {C}^*$-algebra has no nonzero, unital quotients. This implies that the radius of comparison of such a 
$\mathrm {C}^*$-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 
$1$.
