  In this paper we study three aspects of (P(M)/~), the set of Murray-von
Neumann equivalence classes of projections in a von Neumann algebra M. First we
determine the topological structure that (P(M)/~) inherits from the operator
topologies on M. Then we show that there is a version of the center-valued
trace which extends the dimension function, even when M is not sigma-finite.
Finally we prove that (P(M)/~) is a complete lattice, a fact which has an
interesting reformulation in terms of representations.

Operator Algebras;Functional Analysis