The Elliott conjecture for Villadsen algebras of the first type

Details The Elliott conjecture for Villadsen algebras of the first type The Elliott conjecture for Villadsen algebras of the first type

2006-11-02

arxiv.org/abs/math/0611059v1

Mathematics

Operator Algebras

25 pages

Scientific paper

We study the class of simple C*-algebras introduced by Villadsen in his pioneering work on perforated ordered K-theory. We establish six equivalent characterisations of the proper subclass which satisfies the strong form of Elliott's classification conjecture: two C*-algebraic (Z-stability and approximate divisibility), one K-theoretic (strict comparison of positive elements), and three topological (finite decomposition rank, slow dimension growth, and bounded dimension growth). The equivalence of Z-stability and strict comparison constitutes a stably finite version of Kirchberg's characterisation of purely infinite C*-algebras. The other equivalences confirm, for Villadsen's algebras, heretofore conjectural relationships between various notions of good behaviour for nuclear C*-algebras.

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