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Affiliated subspaces and infiniteness of von Neumann algebras

Authors

  • Jan Hamhalter,

    Corresponding author
    1. Czech Technical University in Prague – El. Eng. Department of Mathematics, Technicka 2, 166 27 Prague 6, Czech Republic
    • Czech Technical University in Prague – El. Eng. Department of Mathematics, Technicka 2, 166 27 Prague 6, Czech Republic. Phone: +42 0224353438, Fax: +42 0233339238
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  • Ekaterina Turilova

    1. Czech Technical University in Prague – El. Eng. Department of Mathematics, Technicka 2, 166 27 Prague 6, Czech Republic
    2. Kazan Federal University, Institute of Computational Mathematics and Information Technologies, Kremlevskaya 18, Kazan, Russia. Phone: +7(843)233-71-55, Fax: +7(843)233-71-55
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Abstract

We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi-splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi-splitting subspaces are non-equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi-splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side.

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