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On topologically finite-dimensional simple C*-algebras

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Abstract

We show that, if a simple C*-algebra A is topologically finite-dimensional in a suitable sense, then not only K0(A) has certain good properties, but A is even accessible to Elliott’s classification program. More precisely, we prove the following results:

If A is simple, separable and unital with finite decomposition rank and real rank zero, then K0(A) is weakly unperforated.

If A has finite decomposition rank, real rank zero and the space of extremal tracial states is compact and zero-dimensional, then A has stable rank one and tracial rank zero. As a consequence, if B is another such algebra, and if A and B have isomorphic Elliott invariants and satisfy the Universal coefficients theorem, then they are isomorphic.

In the case where A has finite decomposition rank and the space of extremal tracial states is compact and zero-dimensional, we also give a criterion (in terms of the ordered K0-group) for A to have real rank zero. As a byproduct, we show that there are examples of simple, stably finite and quasidiagonal C*-algebras with infinite decomposition rank.

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Authors and Affiliations

  1. Mathematisches Institut der Universität Münster, Einsteinstr. 62, 48149, Münster

    Wilhelm Winter

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  1. Wilhelm Winter
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Correspondence to Wilhelm Winter.

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Supported by: EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280) and Deutsche Forschungsgemeinschaft (SFB 478).

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Winter, W. On topologically finite-dimensional simple C*-algebras. Math. Ann. 332, 843–878 (2005). https://doi.org/10.1007/s00208-005-0657-z

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