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Lattice isomorphisms between projection lattices of von Neumann algebras

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

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

References

Albeverio, S., Ayupov, S., Kudaybergenov, K. and Djumamuratov, R., ‘Automorphisms of central extensions of type I von Neumann algebras’, Studia Math. 207 (2011) 1–17.CrossRefGoogle Scholar
Dales, H. G., Banach Algebras and Automatic Continuity , London Mathematical Society Monographs, New Series, 24 (New York, The Clarendon Press, 2000).Google Scholar
Dye, H. A., ‘On the geometry of projections in certain operator algebras’, Ann. of Math. (2) 61 (1955) 73–89.CrossRefGoogle Scholar
Feldman, J., ‘Isomorphisms of finite type II rings of operators’, Ann. of Math. (2) 63 (1956) 565–571.CrossRefGoogle Scholar
Fillmore, P. A. and Longstaff, W. E., ‘On isomorphisms of lattices of closed subspaces’, Canad. J. Math. 36 (1984) 820–829.CrossRefGoogle Scholar
Gardner, L. T., ‘On isomorphisms of C ${}^{\ast }$ -algebras’, Amer. J. Math. 87 (1965) 384–396.CrossRefGoogle Scholar
Goodearl, K. R., ‘Von Neumann regular rings: Connections with functional analysis’, Bull. Amer. Math. Soc. (N.S.) 4 (1981) 125–134.CrossRefGoogle Scholar
Halmos, P. R., ‘Two subspaces’, Trans. Amer. Math. Soc. 144 (1969) 381–389.CrossRefGoogle Scholar
Kadison, R. V. and Ringrose, J. R., Fundamentals of the Theory of Operator Algebras , Vol. II (Orlando, Academic Press, 1986).Google Scholar
Kakutani, S. and Mackey, G. W., ‘Ring and lattice characterization of complex Hilbert space’, Bull. Amer. Math. Soc. 52 (1946) 727–733.CrossRefGoogle Scholar
Kaplansky, I., ‘Ring isomorphisms of Banach algebras’, Canad. J. Math. 6 (1954) 374–381.CrossRefGoogle Scholar
Kusraev, A. G., ‘Automorphisms and derivations in an extended complex $f\!$ -algebra’, Sibirsk. Mat. Zh. 47 (2006) 97–107; translation in Sib. Math. J. 47 (2006) 77–85.Google Scholar
Martindale, W. S. III, ‘When are multiplicative mappings additive?’, Proc. Amer. Math. Soc. 21 (1969) 695–698.CrossRefGoogle Scholar
McAsey, M., ‘Spatial implementation of lattice isomorphisms: Selfadjoint and nonselfadjoint operator algebras and operator theory’, Contemp. Math. 120 (1991) 113–115.CrossRefGoogle Scholar
Mori, M., ‘Isometries between projection lattices of von Neumann algebras’, J. Funct. Anal. 276 (2019) 3511–3528.CrossRefGoogle Scholar
Mori, M., ‘Order isomorphisms of operator intervals in von Neumann algebras’, Integral Equations Operator Theory 91 (2019) 11.CrossRefGoogle Scholar
Murray, F. J. and von Neumann, J., ‘On rings of operators’, Ann. of Math. (2) 37 (1936) 116–229.CrossRefGoogle Scholar
Okayasu, T., ‘A structure theorem of automorphisms of von Neumann algebras’, Tohoku Math. J. (2) 20 (1968) 199–206.CrossRefGoogle Scholar
Pisier, G., Tensor Products of C ${}^{\ast }$ -algebras and Operator Spaces (Cambridge, Cambridge University Press, 2020).Google Scholar
Sakai, S., C ${}^{\ast }$ -algebras and W ${}^{\ast }$ -algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, 60 (Springer, New York, 1971).Google Scholar
Segal, I. E., ‘A non-commutative extension of abstract integration’, Ann. of Math. (2) 57 (1953) 401–457.CrossRefGoogle Scholar
von Neumann, J., Continuous Geometry Princeton Mathematical Series, 25 (Princeton, Princeton University Press, 1960).Google Scholar
Yeadon, F. J., ‘Convergence of measurable operators’, Math. Proc. Cambridge Philos. Soc. 74 (1973) 257–268.CrossRefGoogle Scholar