ArticleOriginal scientific text

Title

The Gleason-Kahane-Żelazko theorem and its generalizations

Authors 1

Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., V8W 3P4, Canada

Abstract

This expository article deals with results surrounding the following question: Which pairs of Banach algebras A and B have the property that every unital invertibility preserving linear map from A to B is a Jordan homomorphism?

Bibliography

  1. B. Aupetit, Propriétés Spectrales des Algèbres des Banach, Lecture Notes in Math. 735, Springer, 1979.
  2. M.-D. Choi, D. Hadwin, E. Nordgren, H. Radjavi and P. Rosenthal, On positive linear maps preserving invertibility, J. Funct. Anal. 59 (1984), 462-469.
  3. J. Dieudonné, Sur une généralisation du groupe orthogonal à quatre variables, Arch. Math. (Basel) 1 (1949), 282-287.
  4. M. Eidelheit, On isomorphisms of rings of linear operators, Studia Math. 9 (1940), 97-105.
  5. A. Gleason, A characterization of maximal ideals, J. Analyse Math. 19 (1967), 171-172.
  6. I. N. Herstein, Topics in Ring Theory, University of Chicago Press, Chicago, 1969.
  7. J.-C. Hou, Rank preserving linear maps on ℬ(X), Science in China (Series A) 32 (1989), 929-940.
  8. A. Jafarian and A. R. Sourour, Spectrum preserving linear maps, J. Funct. Anal. 66 (1986), 255-261.
  9. J.-P. Kahane and W. Żelazko, A characterization of maximal ideals in commutative Banach algebras, Studia Math. 29 (1968), 339-343.
  10. I. Kaplansky, Algebraic and Analytic Aspects of Operator Algebras, Amer. Math. Soc., Providence, 1970.
  11. M. Marcus and R. Purves, Linear transformations on algebras of matrices: The invariance of the elementary symmetric functions, Canad. J. Math. 11 (1959), 383-396.
  12. M. Roitman and Y. Sternfeld, When is a linear functional multiplicative?, Trans. Amer. Math. Soc. 267 (1981), 111-124.
  13. B. Russo, Linear mappings of operator algebras, Proc. Amer. Math. Soc. 17 (1966), 1019-1022.
  14. A. R. Sourour, Invertibility preserving linear maps, preprint, 1992.
  15. W. Żelazko, A characterization of multiplicative linear functionals in complex Banach algebras, Studia Math. 30 (1968), 83-85.
Pages:
327-331
Main language of publication
English
Published
1994
Exact and natural sciences