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Linear Preserver Problems

Part of the Frontiers in Mathematics book series (FM)

Abstract

“Linear Preserver Problems” are a very popular research area especially in Linear Algebra, and also in Operator Theory and Functional Analysis. These problems deal with linear maps between algebras that, roughly speaking, preserve certain properties; the goal is to find the form of these maps. This is indeed a rather vague description, and certainly one could explain what is a linear preserver problem in a more precise and systematic manner. But let us instead give a couple of illustrative examples.

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Literature and Comments

  1. [7]
    P. Ara, M. Mathieu, Local multipliers of C*-algebras, Springer, 2003.Google Scholar
  2. [12]
    R. Banning, M. Mathieu, Commutativity preserving mappings on semiprime rings, Comm. Algebra 25 (1997), 247–265.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [14]
    L. B. Beasley, Linear transformations on matrices: The invariance of commuting pairs of matrices, Linear and Multilinear Algebra 6 (1978/79), 179–183.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [18]
    K. I. Beidar, M. Brešar, M.A. Chebotar, Functional identities on upper triangular matrix algebras, J. Math. Sci. (New York) 102 (2000), 4557–4565.zbMATHGoogle Scholar
  5. [19]
    K. I. Beidar, M. Brešar, M.A. Chebotar, Functional identities revised: the fractional and the strong degree, Comm. Algebra 30 (2002), 935–969.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [21]
    K. I. Beidar, M. Brešar, M.A. Chebotar, Y. Fong, Applying functional identities to some linear preserver problems, Pacific J. Math. 204 (2002), 257–271.zbMATHMathSciNetGoogle Scholar
  7. [27]
    K. I. Beidar, S.-C. Chang, M.A. Chebotar, Y. Fong, On functional identities in left ideals of prime rings, Comm. Algebra 28 (2000), 3041–3058.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [36]
    K. I. Beidar, Y.-F. Lin, On surjective linear maps preserving commutativity, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 1023–1040.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [37]
    K. I. Beidar, Y.-F. Lin, Maps characterized by action on Lie zero products, Comm. Algebra 33 (2005), 2697–2703.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [48]
    D. Benkoviç, D. Eremita, Commuting traces and commutativity preserving maps on triangular algebras, J. Algebra 280 (2004), 797–824.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [58]
    M. Brešar, Commuting traces of biadditive mappings, commutativitypreserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), 525–546.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [68]
    M. Brešar, Commutativity preserving maps revisited, Israel J. Math., to appear.Google Scholar
  13. [79]
    M. Brešar, C. R. Miers, Commutativity preserving mappings of von Neumann algebras, Canad. J. Math. 45 (1993), 695–708.zbMATHMathSciNetGoogle Scholar
  14. [82]
    M. Brešar, P. Šemrl, Normal-preserving linear mappings, Canad. Math. Bull. 37 (1994), 306–309.zbMATHMathSciNetGoogle Scholar
  15. [85]
    M. Brešar, P. Šemrl, Elementary operators as Lie homomorphisms or commutativity preservers, Proc. Edinb. Math. Soc. 48 (2005), 37–49.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [86]
    M. Brešar, P. Šemrl, On bilinear maps on matrices with applications to commutativity preservers, J. Algebra 301 (2006), 803–837.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [91]
    M. A. Chebotar, Y. Fong, P.-H. Lee, On maps preserving zeros of the polynomial xyyx*, Linear Algebra Appl. 408 (2005), 230–243.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [94]
    M. A. Chebotar, W.-F. Ke, P.-H. Lee, Maps characterized by action on zero products, Pacific J. Math. 216 (2004), 217–228.zbMATHMathSciNetCrossRefGoogle Scholar
  19. [95]
    M. A. Chebotar, W.-F. Ke, P.-H. Lee, Maps preserving zero Jordan products on Hermitian operators, Illinois J. Math. 49 (2005), 445–452.zbMATHMathSciNetGoogle Scholar
  20. [96]
    M. A. Chebotar, W.-F. Ke, P.-H. Lee, On maps preserving square-zero matrices, J. Algebra 289 (2005), 421–445.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [97]
    M. A. Chebotar, W.-F. Ke, P.-H. Lee, L.-S. Shiao, On maps preserving products, Canad. Math. Bull. 48 (2005), 355–369.MathSciNetGoogle Scholar
  22. [98]
    M. A. Chebotar, W.-F. Ke, P.-H. Lee, R.-B. Zhang, On maps preserving zero Jordan products, Monatshefte Math. 149 (2006), 91–101.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [100]
    M. D. Choi, A.A. Jafarian, H. Radjavi, Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987), 227–242.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [101]
    J. Cui, J. Hou, Linear maps preserving elements annihilated by the polynomial XYYX†, Studia Math. 174 (2006), 183–199.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [131]
    C. M. Kunicki, R.D. Hill, Normal-preserving linear transformations, Linear Algebra Appl. 170 (1992), 107–115.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [145]
    Y.-F. Lin, Commutativity-preserving maps on Lie ideals of prime algebras, Linear Algebra Appl. 371 (2003), 361–368.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [168]
    C. R. Miers, Commutativity preserving maps of factors, Canad. J. Math. 40 (1988), 248–256.zbMATHMathSciNetGoogle Scholar
  28. [176]
    M. Omladiç, On operators preserving commutativity, J. Funct. Anal. 66 (1986), 105–122.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [177]
    M. Omladič, H. Radjavi, P. Šemrl, Preserving commutativity, J. Pure Appl. Algebra 156 (2001), 309–328.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [179]
    S. Pierce, W. Watkins, Invariants of linear maps on matrix algebras, Linear and Multilinear Algebra 6 (1978/79), 185–200.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [180]
    V. P. Platonov, D. Ž. Djoković, Linear preserver problems and algebraic groups, Math. Ann. 303 (1995), 165–184.zbMATHCrossRefMathSciNetGoogle Scholar
  32. [181]
    V. P. Platonov, D. Ž. Djoković, Subgroups of GL(n 2, ℂ) containing PSU(n), Trans. Amer. Math. Soc. 348 (1996), 141–152.zbMATHCrossRefMathSciNetGoogle Scholar
  33. [196]
    W. Watkins, Linear maps that preserve commuting pairs of matrices, Linear Algebra Appl. 14 (1976), 29–35.zbMATHCrossRefMathSciNetGoogle Scholar
  34. [201]
    L. Zhao, J. Hou, Jordan zero-product preserving additive maps on operator algebras, J. Math. Anal. Appl. 314 (2006), 689–700.zbMATHCrossRefMathSciNetGoogle Scholar

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© Birkhäuser Verlag 2007

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