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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras

Nathanial P. Brown1 / Francesc Perera2 / Andrew S. Toms3

1Department of Mathematics, Penn State University, State College, PA 16802, USA. e-mail: nbrown@math.psu.edu

2Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain. e-mail: perera@mat.uab.cat

3Department of Mathematics and Statistics, York University, 4700 Keele St., Toronto, Ontario, M3J 1P3 Canada. e-mail: atoms@mathstat.yorku.ca

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2008, Issue 621, Pages 191–211, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2008.062, July 2008

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We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C*-algebras. In particular, our results apply to the largest class of simple C*-algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliott's classification program, proving that the usual form of the Elliott conjecture is equivalent, among -stable algebras, to a conjecture which is in general substantially weaker and for which there are no known counterexamples. Second and third, it resolves, for the class of algebras above, two conjectures of Blackadar and Handelman concerning the basic structure of dimension functions on C*-algebras. We also prove in passing that the Cuntz-Pedersen semigroup is recovered functorially from the Elliott invariant for a large class of simple unital C*-algebras.

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