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Lattice isomorphisms between projection lattices of von Neumann algebras

Part of: Special classes of linear operators Selfadjoint operator algebras Geometric closure systems Homological methods

Published online by Cambridge University Press:  13 November 2020

Michiya Mori
Michiya Mori*
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914, Japan; E-mail: mmori@ms.u-tokyo.ac.jp

Abstract

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Generalizing von Neumann’s result on type II $_1$ von Neumann algebras, I characterise lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, I give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.

MSC classification

Primary: 46L10: General theory of von Neumann algebras
Secondary: 16E50: von Neumann regular rings and generalizations 47B49: Transformers, preservers (operators on spaces of operators) 51D25: Lattices of subspaces
Type
Analysis
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e49
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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