The Cuntz semigroup of some spaces of dimension at most two

2007/11/28 by Leonel Robert, Robert, Leonel · 1 citation
Mathematics · #46L08 #46L35 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.0711.4396

openalex publication_date 2007/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is shown that the Cuntz semigroup of a space with dimension at most two, and with second cohomology of its compact subsets equal to zero, is isomorphic to the ordered semigroup of lower semicontinuous functions on the space with values in the natural numbers with the infinity adjoined. This computation is then used to obtain the Cuntz semigroup of all compact surfaces. A converse to the first computation is also proven: if the Cuntz semigroup of a separable C*-algebra is isomorphic to the lower semicontinuous functions on a topological space with values in the extended natural numbers, then the C*-algebra is commutative up to stability, and its spectrum satisfies the dimensional and cohomological conditions mentioned above.

Cited by

• Self-adjoint operators in Z-stable C^*-algebras with prescribed spectral data

Related

• Pullbacks, C(X)-algebras, and their Cuntz semigroup
• Covering dimension of Cuntz semigroups
• Covering dimension of Cuntz semigroups II
• Recovering the Elliott invariant from the Cuntz semigroup
• The Cuntz semigroup of continuous fields