RANKS OF SOFT OPERATORS IN NOWHERE SCATTERED $\mathrm {C}^*$-ALGEBRAS
Abstract 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 $1$ .
Authors
- Hannes Thiel (ORCID: https://orcid.org/0000-0003-0388-6495)
- M. Ali Asadi-Vasfi (ORCID: https://orcid.org/0009-0007-0967-1491)
- Eduard Vilalta (ORCID: https://orcid.org/0000-0002-1980-9323)
Institutions
- University of Toronto (CA)
- Chalmers University of Technology (SE)
- University of Gothenburg (SE)
- Fields Institute for Research in Mathematical Sciences (CA)
Publication Details
- Journal
- Journal of the Institute of Mathematics of Jussieu
- Published
- 2024-12-20
- DOI
- https://doi.org/10.1017/s1474748024000318
- Citations
- 2
- Primary Topic
- Advanced Topics in Algebra
- Type
- article
- Field-Weighted Citation Impact
- 2.54
Funders
- Deutsche Forschungsgemeinschaft
- Generalitat de Catalunya
- Knut och Alice Wallenbergs Stiftelse
- Akademie Věd České Republiky
- Westfälische Wilhelms-Universität Münster
- Division of Mathematical Sciences