The Elliott conjecture for Villadsen algebras of the first type

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. Crown Copyright © 2008 Published by Elsevier Inc. All rights reserved.