Recovering the Elliott invariant from the Cuntz semigroup

Ramon Antoine, Marius Dadarlat, Francesc Perera, Luis Santiago

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 languageEnglish
Pages (from-to)2907-2922
Number of pages16
JournalTransactions of the American Mathematical Society
Volume366
Issue number6
Early online date13 Feb 2014
DOIs
Publication statusPublished - 2014

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Semigroup
Invariant
Functor
Stable Rank
Lower Semicontinuous Function
Tensor Product
C*-algebra
Continuous Function
Circle
Projection
Algebra

Cite this

Recovering the Elliott invariant from the Cuntz semigroup. / Antoine, Ramon; Dadarlat, Marius; Perera, Francesc; Santiago, Luis.

In: Transactions of the American Mathematical Society, Vol. 366, No. 6, 2014, p. 2907-2922.

Research output: Contribution to journalArticle

Antoine, Ramon ; Dadarlat, Marius ; Perera, Francesc ; Santiago, Luis. / Recovering the Elliott invariant from the Cuntz semigroup. In: Transactions of the American Mathematical Society. 2014 ; Vol. 366, No. 6. pp. 2907-2922.
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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.",
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note = "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.",
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