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RANKS OF SOFT OPERATORS IN NOWHERE SCATTERED
$\mathrm {C}^*$-ALGEBRAS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 24 / Issue 2 / March 2025
- Published online by Cambridge University Press:
- 20 December 2024, pp. 371-410
- Print publication:
- March 2025
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- Article
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- You have access
- Open access
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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
$1$.
