Maps preserving \(\varvec{A}^{\varvec{*}}\varvec{A+AA}^{\varvec{*}}\) on \(\varvec{C}^{\varvec{*}}\)-algebras

  • Ali TaghaviEmail author


Let \(\mathcal {A}\) be a \(C^*\)-algebra of real-rank zero and \(\mathcal {B}\) be a \(C^{*}\)-algebra with unit I. It is shown that the mapping \(\Phi : {{\mathcal {A}}}\longrightarrow {{\mathcal {B}}}\) which preserves arithmetic mean and satisfies
$$\begin{aligned} \Phi (A^{*}A)=\frac{\Phi (A)^{*}\Phi (A)+\Phi (A)\Phi (A)^{*}}{2}, \end{aligned}$$
for all normal elements \(A\in \mathcal {A}\), is an \({\mathbb {R}}\)-linear continuous Jordan \(*\)-homomorphism provided that \(0\in \mathrm{Ran}\ \Phi \). Also, \(\Phi \) is the sum of a linear Jordan \(*\)-homomorphism and a conjugate-linear Jordan \(*\)-homomorphism. This result also presents an application of maps which preserve the square absolute value.


\(C^*\)-algebra \({\mathbb {C}}\)-linear \(\mathbb C\)-antilinear homomorphism linear preserver problem real rank zero 

2010 Mathematics Subject Classification

47B48 46L10 



The author is thankful to the anonymous reviewer(s) for their careful reading of this paper and for their valuable suggestions in rewriting the paper in the present form.


  1. 1.
    Brešar M, Commuting traces of biadditive mappings, commutativity-preserving, mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993) 525–546MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brown L G and Pedersen G, \(C^*\)-algebras of real rank zero, J. Funct. Anal. 9 (1991) 131–149MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chan G H and Lim M H, Linear preservers on powers of matrices, Linear Algebra Appl. 162–164 (1992) 615–626MathSciNetCrossRefGoogle Scholar
  4. 4.
    Choi M D, Jafarian A A and Radjavi H, Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987) 227–241MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cui J and Li C K, Maps preserving product \(XY - Y X^{*}\) on factor von Neumann algebras, Linear Algebra Appl. 431 (2009) 833–842MathSciNetCrossRefGoogle Scholar
  6. 6.
    Douglas R G, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966) 413–416MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jafarian A A and Sourour A R, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986) 255–261MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ji P and Liu Z, Additivity of Jordan maps on standard Jordan operator algebras, Linear Algebra Appl. 430 (2009) 335–343MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li C K and Tsing N K, Linear preserver problems: A brief introduction and some special techniques, Linear Algebra Appl. 162–164 (1992) 217–235MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li C, Lu F and Fang X, Nonlinear mappings preserving product \(XY+YX^{*}\) on factor von Neumann algebras, Linear Algebra Appl. 438 (2013) 2339–2345MathSciNetCrossRefGoogle Scholar
  11. 11.
    Liu L and Ji G X, Maps preserving product \(X^{*}Y+YX^{*}\) on factor von Neumann algebras, Linear and Multilinear Algebra 59 (2011) 951–955MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lu F, Jordan triple maps, Linear Algebra Appl. 375 (2003) 311–317MathSciNetCrossRefGoogle Scholar
  13. 13.
    Martindale W S III, When are multiplicative mappings additive? Proc. Amer. Math. Soc. 21 (1969) 695–698MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mires C R, Lie isomorphisms of factors, Trans. Amer. Math. Soc. 147 (1970) 5–63MathSciNetGoogle Scholar
  15. 15.
    Molnár L, A condition for a subspace of \(B(H)\) to be an ideal, Linear Algebra Appl. 235 (1996) 229–234MathSciNetCrossRefGoogle Scholar
  16. 16.
    Molnár L, Two characterisations of additive \(\ast \)-automorphism of \(B(H)\), Bull. Aust. Math. Soc. 53 (1996) 391–400MathSciNetCrossRefGoogle Scholar
  17. 17.
    Molnár L, Multiplicative Jordan triple isomorphisms on the self-adjoint elements of von Neuman algebras, Linear Algebra Appl. 419 (2006) 586–600MathSciNetCrossRefGoogle Scholar
  18. 18.
    Molnár L, Maps preserving the harmonic mean or the parallel sum of positive operators, Linear Algebra and its Applications 430 (2009) 3058–3065MathSciNetCrossRefGoogle Scholar
  19. 19.
    Omladič M, On operators preserving commutativity, J. Funct. Anal. 66 (1986) 105–122MathSciNetCrossRefGoogle Scholar
  20. 20.
    Omladič M and Šemrl P, Linear mappings that preserve potent operators, Proc. Amer. Math. Soc. 123 (1995) 1069–1074MathSciNetCrossRefGoogle Scholar
  21. 21.
    Qi X and Hou J, Additivity of Lie multiplicative maps on triangular algebras, Linear and Multilinear Algebra 59 (2011) 391–397MathSciNetCrossRefGoogle Scholar
  22. 22.
    Radjabalipour M, Additive mappings on von Neumann algebras preserving absolute values, Linear Algebra Appl. 368 (2003) 229–241MathSciNetCrossRefGoogle Scholar
  23. 23.
    Radjabalipour M, Seddighi K and Taghavi Y, Additive mappings on operator algebras preserving absolute values, Linear Algebra Appl. 327 (2001) 197–206MathSciNetCrossRefGoogle Scholar
  24. 24.
    Śemrl P, Linear mappings preserving square-zero matrices, Bull. Austral. Math. Soc. 48 (1993) 365–370MathSciNetCrossRefGoogle Scholar
  25. 25.
    Šemrl P, Quadratic functionals and Jordan \(*\)-derivations, Studia Math. 97 (1991) 157–165MathSciNetCrossRefGoogle Scholar
  26. 26.
    Taghavi A, Additive mapping on \(C^{*}\)-algebras preserving absolute values, Linear and Multilinear Algebra, 60 (2012), 33–38MathSciNetCrossRefGoogle Scholar
  27. 27.
    Taghavi A and Hosseinzadeh R, Mappings on \(C^*\)-algebras preserving the sum of absolute values, Linear and Multilinear Algebra 66(2) (2018) 217–223MathSciNetCrossRefGoogle Scholar
  28. 28.
    Taghavi A, Darvish V and Rohi H, Additivity of maps preserving products \(AP\pm PA^{*}\), Mathematica Slovaca. 67(1) (2017) 213–220MathSciNetzbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of MazandaranBabolsarIran

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