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.