Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-06-01T21:32:32.348Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Beasley, L. B., Linear transformations on matrices: The invariance of commuting pairs of matrices, Linear and Multilinear Algebra 6(1978), 179–183.Google Scholar
2. Bresar, M, Jordan mappings of semiprime rings, J. Algebra 127(1989), 218–228.Google Scholar
3. Bresar, M, Jordan mappings of semiprime rings II, Bull. Austral. Math. Soc., to appear.Google Scholar
4. Bresar, M, Centralizing mappings and derivations in prime rings, J. Algebra, to appear.Google Scholar
5. Bresar, M, Centralizing mappings on von Neumann algebras, Proc. Amer. Math. Soc. 111(1991), 501–510.Google Scholar
6. Bresar, M, Commuting traces of biadditive mappings, commutativity preserving mappings, and Lie mappings, Trans. Amer. Math. Soc, to appear.Google Scholar
7. Chan, G. H. and Lim, M. H., Linear transformations on symmetric matrices that preserve commutativity, Linear Algebra Appl. 47(1982), 11–22.Google Scholar
8. Choi, M. D., Jafarian, A. A. and Radjavi, H., Linear maps preserving commutativity, Linear Algebra Appl. 87(1987), 227–242.Google Scholar
9. Kadison, R. V. and Ringrose, J. R., Fundamentals of the thoery of operator algebras, 1, Academic Press, London, 1983; 2, Academic Press, London, 1986.Google Scholar
10. Miers, C. R., Lie homomorphisms of operator algebras, Pacific J. Math. 38(1971), 717–735.Google Scholar
11. Miers, C. R., ClosedLie ideals in operator algebras, Canadian J. Math. XXXIII(1981), 1271–1278.Google Scholar
12. Miers, C. R., Commutativity preserving mappings of factors, Canadian J. Math. 40(1988), 248–256.Google Scholar
13. Omladic, M., On operators preserving commutativity, J. Funct. Anal. 66(1986), 105–122.Google Scholar
14. Pierce, S. and Watkins, W., Invariants of linear maps on matrix algebras, Linear and Multilinear Algebra 6(1978), 185–200.Google Scholar
15. Radjavi, H., Commutativity preserving operators on symmetric matrices, Linear Algebra Appl. 61(1984), 219–224.Google Scholar
16. Watkins, W., Linear maps that preserve commuting pairs of matrices, Linear Algebra Appl. 14(1976), 29– 35.Google Scholar
17. Okaysu, T., A structure theorem of automorphisms of von Neumann algebras, Tôhoku Math. Journal 20 (1968), 199–206.Google Scholar
18. Putnam, C. R., Commutation Properties of Hilbert Space Operators and Related Topics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36, Springer-Verlag New York Inc., 1967.Google Scholar