The Cuntz semigroup of continuous fields

TL;DR

This paper characterizes the Cuntz semigroup of continuous fields of C*-algebras over one-dimensional spaces with specific fiber properties, linking it to global sections and sheaf invariants, advancing classification methods.

Contribution

It provides a new description of the Cuntz semigroup for certain continuous fields, connecting it to topological and sheaf-theoretic invariants for classification.

Findings

Cuntz semigroup described via global sections of a topological space.

Classification depends on the sheaf induced by the Murray-von Neumann semigroup.

Applicable to fibers with stable rank one and trivial K_1, over one-dimensional spaces.

Abstract

In this paper we describe the Cuntz semigroup of continuous fields of C$^*$-algebras over one dimensional spaces whose fibers have stable rank one and trivial $K_1$ for each closed, two-sided ideal. This is done in terms of the semigroup of global sections on a certain topological space built out of the Cuntz semigroups of the fibers of the continuous field. When the fibers have furthermore real rank zero, and taking into account the action of the space, our description yields that the Cuntz semigroup is a classifying invariant if and only if so is the sheaf induced by the Murray-von Neumann semigroup.