Recovering the Elliott invariant from the Cuntz semigroup

Abstract

Let A A be a simple, separable C ∗ ^* -algebra of stable rank one. We prove that the Cuntz semigroup of C ( T , A ) \mathrm {C}(\mathbb {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 A ). This result has two consequences. First, specializing to the case that A A is simple, finite, separable and Z \mathcal Z -stable, this yields a description of the Cuntz semigroup of C ( T , A ) \mathrm {C}(\mathbb {T},A) in terms of the Elliott invariant of A 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.