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On the dimension theory of von Neumann algebras

David Sherman
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Abstract

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.

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David Sherman. 2005-03-31. On the dimension theory of von Neumann algebras. https://arxiv.org/abs/math/0503747

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