Types of von Neumann Algebras

Lars Tuset
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Lars Tuset: Oslo Metropolitan University, Department of Computer Science

Chapter Chapter 7 in Analysis and Quantum Groups, 2022, pp 193-223 from Springer

Abstract: Abstract The classification of von Neumann algebras requires a good understanding of their orthogonal projections. We show that they form a complete lattice under the usual order given by positivity, so the supremum and infinum of any family of them belong to the von Neumann algebra W. We say two orthogonal projections p, q ∈ W are equivalent if the corresponding reduced representations of the identity representation of W′ associated to them are unitarily equivalent. This means that there is w ∈ W such that p = ww∗ and q = w∗w. The set D(W) of equivalence classes of such projections is then partially ordered, then given by comparing representatives, and it is a partial semigroup under addition of representatives whenever these are mutually orthogonal. The partial order on D(W) becomes a total order when W is a factor.

Date: 2022
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DOI: 10.1007/978-3-031-07246-8_7

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