  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 \otimes A isomorphic
to 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?
