The Cuntz semigroup of continuous functions into certain simple C*-algebras

TL;DR

This paper computes the Cuntz semigroup for certain continuous function C*-algebras, relating it to Murray-von Neumann semigroups, advancing classification methods for non-simple C*-algebras.

Contribution

It provides explicit computations of the Cuntz semigroup for C_0(X,A) where A is a unital, simple, Z-stable ASH algebra, linking it to the Elliott invariant.

Findings

Cuntz semigroup described via Murray-von Neumann semigroups of C(K,A)

Elliott invariant is functorially equivalent to the Cuntz semigroup of C(T,A)

Advances classification of non-simple C*-algebras

Abstract

This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C_0(X,A), where A is a unital, simple, Z-stable ASH algebra. The computations describe the Cuntz semigroup in terms of Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(T,A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of well-behaved non-simple C*-algebras.