### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 191-211 |

Number of pages | 21 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 621 |

DOIs | |

State | Published - Aug 1 2008 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal fur die Reine und Angewandte Mathematik*, (621), 191-211. https://doi.org/10.1515/CRELLE.2008.062

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*Journal fur die Reine und Angewandte Mathematik*, no. 621, pp. 191-211. https://doi.org/10.1515/CRELLE.2008.062

**The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras.** / Brown, Nathanial P.; Perera, Francesc; Toms, Andrew S.

Research output: Contribution to journal › Article

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.

PY - 2008/8/1

Y1 - 2008/8/1

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.

UR - http://www.scopus.com/inward/record.url?scp=46649111900&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=46649111900&partnerID=8YFLogxK

U2 - 10.1515/CRELLE.2008.062

DO - 10.1515/CRELLE.2008.062

M3 - Article

AN - SCOPUS:46649111900

SP - 191

EP - 211

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 621

ER -