  We examine the ranks of operators in semi-finite C*-algebras as measured by
their densely defined lower semicontinuous traces. We first prove that a unital
simple C*-algebra whose extreme tracial boundary is nonempty and finite
contains positive operators of every possible rank, independent of the property
of strict comparison. We then turn to nonunital simple algebras and establish
criteria that imply that the Cuntz semigroup is recovered functorially from the
Murray-von Neumann semigroup and the space of densely defined lower
semicontinuous traces. Finally, we prove that these criteria are satisfied by
not-necessarily-unital approximately subhomogeneous algebras of slow dimension
growth.
