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Recovering the Elliott invariant from the Cuntz semigroup


Authors: Ramon Antoine, Marius Dadarlat, Francesc Perera and Luis Santiago
Journal: Trans. Amer. Math. Soc. 366 (2014), 2907-2922
MSC (2010): Primary 46L05, 46L35, 46L80, 19K14
DOI: https://doi.org/10.1090/S0002-9947-2014-05833-9
Published electronically: February 13, 2014
MathSciNet review: 3180735
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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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Additional Information

Ramon Antoine
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Email: ramon@mat.uab.cat

Marius Dadarlat
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: mdd@math.purdue.edu

Francesc Perera
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Email: perera@mat.uab.at

Luis Santiago
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Address at time of publication: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: santiago@mat.uab.cat, luiss@uoregon.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05833-9
Received by editor(s): September 27, 2011
Received by editor(s) in revised form: June 7, 2012
Published electronically: February 13, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.