Ranks of operators in simple C ∗ - algebras ✩

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? © 2010 Elsevier Inc. All rights reserved.