Recovering the Elliott invariant from the Cuntz semigroup

TL;DR

This paper demonstrates how the Cuntz semigroup of a certain class of C*-algebras over the circle can be derived from the Elliott invariant, linking two important classification tools in operator algebras.

Contribution

It establishes a method to recover the Elliott invariant from the Cuntz semigroup for a class of C*-algebras, connecting these invariants in a novel way.

Findings

Cuntz semigroup of $ ext{C}( ext{T},A)$ is determined by projections and lower semicontinuous functions.

For $ ext{A}$ simple, finite, separable, and $ ext{Z}$-stable, the Cuntz semigroup relates to the Elliott invariant.

The Elliott functor and the Cuntz semigroup functor are naturally equivalent for these algebras.

Abstract

Let $A$ be a simple, separable C$^*$-algebra of stable rank one. We prove that the Cuntz semigroup of $\CC(\T,A)$ is determined by its Murray-von Neumann semigroup of projections and a certain semigroup of lower semicontinuous functions (with values in the Cuntz semigroup of $A$). This result has two consequences. First, specializing to the case that $A$ is simple, finite, separable and $\mathcal Z$-stable, this yields a description of the Cuntz semigroup of $\CC(\T,A)$ in terms of the Elliott invariant of $A$. Second, suitably interpreted, it shows that the Elliott functor and the functor defined by the Cuntz semigroup of the tensor product with the algebra of continuous functions on the circle are naturally equivalent.