The Cuntz semigroup of some spaces of dimension at most two

TL;DR

This paper computes the Cuntz semigroup for spaces of dimension at most two with specific cohomological properties, and applies this to classify the semigroup of all compact surfaces, linking it to the algebraic structure of associated C*-algebras.

Contribution

It provides an explicit description of the Cuntz semigroup for certain low-dimensional spaces and establishes a converse characterization relating the semigroup to the space's topology and algebraic properties.

Findings

Cuntz semigroup is isomorphic to lower semicontinuous functions for specified spaces.

Computed the Cuntz semigroup for all compact surfaces.

Characterized when a C*-algebra's semigroup corresponds to such topological spaces.

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.