Recovering the Elliott invariant from the Cuntz semigroup
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- by Ramon Antoine, Marius Dadarlat, Francesc Perera and Luis Santiago PDF
- Trans. Amer. Math. Soc. 366 (2014), 2907-2922 Request permission
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.References
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Additional Information
- Ramon Antoine
- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
- MR Author ID: 629811
- Email: ramon@mat.uab.cat
- Marius Dadarlat
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 53925
- Email: mdd@math.purdue.edu
- Francesc Perera
- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
- MR Author ID: 620835
- 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
- Received by editor(s): September 27, 2011
- Received by editor(s) in revised form: June 7, 2012
- Published electronically: February 13, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3180735