Abstract
Let $ A$ be a simple, separable C$ ^*$-algebra of stable rank one. We prove that the Cuntz semigroup of $ \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$). 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 $ \mathrm {C}(\mathbb{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.
| Original language | English |
|---|---|
| Pages (from-to) | 2907-2922 |
| Number of pages | 16 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 366 |
| Issue number | 6 |
| Early online date | 13 Feb 2014 |
| DOIs | |
| Publication status | Published - 2014 |
Bibliographical note
This work was carried out at the Centre de Recerca Matem`atica (Bellaterra) duringthe programme “The Cuntz Semigroup and the Classification of C∗-algebras”
in 2011. We gratefully acknowledge the support and hospitality extended to us.
It is also a pleasure to thank N. Brown, I. Hirshberg, N. C. Phillips and H. Thiel
for interesting discussions concerning the subject matter of this paper. Also, we
would like to thank the referee for a number of helpful comments. The first, third
and fourth authors were partially supported by an MEC-DGESIC grant (Spain)
through Project MTM2008-06201-C02-01/MTM, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The second author was partially supported by NSF grant #DMS–1101305.