Abstract
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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