Isometries between projection lattices of von Neumann algebras

We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan ⁎-isomorphisms. In particular, we prove that two von Neumann algebras without type I1 direct summands are Jordan ⁎-isomorphic if and only if t...

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Bibliographic Details
Published inJournal of functional analysis Vol. 276; no. 11; pp. 3511 - 3528
Main Author Mori, Michiya
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.06.2019
Subjects
Grassmann space
Isometry
Projection
von Neumann algebra
secondary
Projection
Isometry
Grassmann space
von Neumann algebra
primary
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Summary:We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan ⁎-isomorphisms. In particular, we prove that two von Neumann algebras without type I1 direct summands are Jordan ⁎-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Gehér and P. Šemrl, which is a generalization of Wigner's theorem.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2018.10.011