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Commutativity Preserving Mappings of Von Neumann Algebras

Published online by Cambridge University Press:  20 November 2018

Matej Brešar  and
C. Robert Miers
Matej Brešar
Affiliation:
University ofMaribor, PF Koroska 160, Y-62000 Maribor, Slovenia
C. Robert Miers
Affiliation:
Department of Mathematics and Statistics, P.O. Box 3045, University of Victoria, Victoria, British Columbia V8W 3P4
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Abstract

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A map θ: M —> N where M and N are rings is said to preserve commutativity in both directions if the elements a,b ∊ M commute if and only if θ(a) and θ(b) commute. In this paper we show that if M and N are von Neumann algebras with no central summands of type I1 or I2 and θ is a bijective additive map which preserves commutativity in both directions then θ(x) = cφ(x) +f(x) where c is an invertible element in ZN, the center of N, φ M —> N is a Jordan isomorphism of M onto N, and f is an additive map of M into ZN.

Keywords

46L1016W10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 4 , 01 August 1993 , pp. 695 - 708
Copyright
Copyright © Canadian Mathematical Society 1993

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