On the dimension theory of von Neumann algebras

Abstract

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