We prove that the Cuntz semigroup is recovered functorially from the Elliott
invariant for a large class of C*-algebras. In particular, our results apply to
the largest class of simple C*-algebras for which K-theoretic classification
can be hoped for. This work has three significant consequences. First, it
provides new conceptual insight into Elliott's classification program, proving
that the usual form of the Elliott conjecture is equivalent, among Z-stable
algebras, to a conjecture which is in general substantially weaker and for
which there are no known counterexamples. Second and third, it resolves, for
the class of algebras above, two conjectures of Blackadar and Handelman
concerning the basic structure of dimension functions on C*-algebras. We also
prove in passing that the Cuntz-Pedersen semigroup is recovered functorially
from the Elliott invariant for a large class of unital simple C*-algebras.Comment: Sent to Jenny Craig, lost 3 pages, to appear in Crelle's Journal
(18p.