  The cone of lower semicontinuous traces is studied with a view to its use as
an invariant. Its properties include compactness, Hausdorffness, and continuity
with respect to inductive limits. A suitable notion of dual cone is given. The
cone of lower semicontinuous 2-quasitraces on a (non-exact) C*-algebra is
considered as well. These results are applied to the study of the Cuntz
semigroup. It is shown that if a C*-algebra absorbs the Jiang-Su algebra, then
the subsemigroup of its Cuntz semigroup consisting of the purely non-compact
elements is isomorphic to the dual cone of the cone of lower semicontinuous
2-quasitraces. This yields a computation of the Cuntz semigroup for the
following two classes of C*-algebras: C*-algebras that absorb the Jiang-Su
algebra and have no non-zero simple subquotients, and simple C*-algebras that
absorb the Jiang-Su algebra.
