  We show that positive elements of a Pedersen ideal of a tensor product can be
approximated in a particularly strong sense by sums of tensor products of
positive elements. This has a range of applications to the structure of tracial
cones and related topics, such as the Cuntz-Pedersen space or the Cuntz
semigroup. For example, we determine the cone of lower semicontinuous traces of
a tensor product in terms of the traces of the tensor factors, in an arbitrary
C*-tensor norm. We show that the positive elements of a Pedersen ideal are
sometimes stable under Cuntz equivalence. We generalize a result of Pedersen's
by showing that certain classes of completely positive maps take a Pedersen
ideal into a Pedersen ideal. We provide theorems that in many cases compute the
Cuntz semigroup of a tensor product.
