TY - JOUR

T1 - The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras

AU - Brown, Nathanial P.

AU - Perera, Francesc

AU - Toms, Andrew S.

N1 - Funding Information:
N.B. was partially supported by DMS-0554870, F.P. by the DGI MEC-FEDER through Project MTM2005-00934, and the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. A.T. was supported by an NSERC Discovery Grant.

PY - 2008/8

Y1 - 2008/8

N2 - 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.

AB - 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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U2 - 10.1515/CRELLE.2008.062

DO - 10.1515/CRELLE.2008.062

M3 - Article

AN - SCOPUS:46649111900

SN - 0075-4102

SP - 191

EP - 211

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

IS - 621

ER -