Recovering the Elliott invariant from the Cuntz semigroup

TY - JOUR

T1 - Recovering the Elliott invariant from the Cuntz semigroup

AU - Antoine, Ramon

AU - Dadarlat, Marius

AU - Perera, Francesc

AU - Santiago, Luis

N1 - This work was carried out at the Centre de Recerca Matem`atica (Bellaterra) during the 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.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

U2 - 10.1090/S0002-9947-2014-05833-9

DO - 10.1090/S0002-9947-2014-05833-9

M3 - Article

VL - 366

SP - 2907

EP - 2922

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 6

ER -