On the dimension theory of von Neumann algebras - Sherman, David (MATHEMATICA SCANDINAVICA, 2007)
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 σ-finite. Finally we prove that (P(M)/) is a complete lattice, a fact which has an interesting reformulation in terms of representations.