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 -