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Von Neumann Algebras and Hilbert Quantales

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Abstract

When A is a von Neumann algebra, the set of all weakly closed linear subspaces forms a Gelfand quantale, Maxw A. We prove that Maxw A is a von Neumann quantale for all von Neumann algebras A. The natural morphism from Maxw A to the Hilbert quantale on the lattice of weakly closed right ideals of A is, in general, not an isomorphism. However, when A is a von Neumann factor, its restriction to right-sided elements is an isomorphism and this leads to a new characterization of von Neumann factors.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, North York, Ontario, Canada, M3J 1P3 (e-mail

    J. Wick Pelletier

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  1. J. Wick Pelletier
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Wick Pelletier, J. Von Neumann Algebras and Hilbert Quantales. Applied Categorical Structures 5, 249–264 (1997). https://doi.org/10.1023/A:1008605720422

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  • DOI: https://doi.org/10.1023/A:1008605720422

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