Ranks of operators in simple [formula omitted]-algebras

Abstract

Let A be a unital simple separable C ∗ -algebra with strict comparison of positive elements. We prove that the Cuntz semigroup of A is recovered functorially from the Murray–von Neumann semigroup and the tracial state space T ( A ) whenever the extreme boundary of T ( A ) is compact and of finite covering dimension. Combined with a result of Winter, we obtain Z ⊗ A ≅ A whenever A moreover has locally finite decomposition rank. As a corollary, we confirm Elliott's classification conjecture under reasonably general hypotheses which, notably, do not require any inductive limit structure. These results all stem from our investigation of a basic question: what are the possible ranks of operators in a unital simple C ∗ -algebra?