Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations

Behrooz Fadaee 1 , Kamal Fallahi 2  and Hoger Ghahramani 1
  • 1 Department of Mathematics, University of Kurdistan, P. O. Box 416, Sanandaj, Iran
  • 2 Department of Mathematics, Payam Noor University of Technology, P.O. Box 19395-3697, Tehran, Iran
Behrooz Fadaee, Kamal Fallahi and Hoger Ghahramani

Abstract

Let 𝓐 be a ⋆-algebra, δ : 𝓐 → 𝓐 be a linear map, and z ∈ 𝓐 be fixed. We consider the condition that δ satisfies (y) + δ(x)y = δ(z) (xδ(y) + δ(x)y = δ(z)) whenever xy = z (xy = z), and under several conditions on 𝓐, δ and z we characterize the structure of δ. In particular, we prove that if 𝓐 is a Banach ⋆-algebra, δ is a continuous linear map, and z is a left (right) separating point of 𝓐, then δ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map δ satisfying the above conditions with z = 0 on two classes of ⋆-algebras: zero product determined algebras and standard operator algebras.

  • [1]

    Alaminos, J.—Brešar, M.—Extremera, J.—Villena, A. R.: Maps preserving zero products, Studia Math. 193 (2009), 131–159.

    • Crossref
    • Export Citation
  • [2]

    An, R.—Hou, J.: Characterizations of derivations on triangular rings: additive maps derivable at idempotents, Linear Algebra Appl. 431 (2009), 1070–1080.

    • Crossref
    • Export Citation
  • [3]

    Brešar, M.: Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinb. Sect. A. 137 (2007), 9–21.

    • Crossref
    • Export Citation
  • [4]

    Brešar, M.—Grašić, M.—Ortega, J. S.: Zero product determined matrix algebras, Linear Algebra Appl. 430 (2009), 1486–1498.

    • Crossref
    • Export Citation
  • [5]

    Brešar, M.: Multiplication algebra and maps determined by zero products, Linear Multilinear Algebra 60 (2012), 763–768.

    • Crossref
    • Export Citation
  • [6]

    Burgos, M.—Ortega, J. S.: On mappings preserving zero products, Linear Multilinear Algebra 61 (2013), 323–335.

    • Crossref
    • Export Citation
  • [7]

    Chebotar, M. A.—Ke, W.-F.—Lee, P.-H.: Maps characterized by action on zero products, Pacific. J. Math. 216 (2004), 217–228.

    • Crossref
    • Export Citation
  • [8]

    Chuang, C. L.—Lee, T. K.: Derivations modulo elementary operators, J. Algebra 338 (2011), 56–70.

    • Crossref
    • Export Citation
  • [9]

    Ghahramani, H.: Additive mappings derivable at nontrivial idempotents on Banach algebras, Linear Multilinear Algebra 60 (2012), 725–742.

    • Crossref
    • Export Citation
  • [10]

    Ghahramani, H.: On rings determined by zero products, J. Algebra Appl. 12 (2013), 1–15.

  • [11]

    Ghahramani, H.: Additive maps on some operator algebras behaving like (α, β)-derivations or generalized (α, β)-derivations at zero-product elements, Acta Math. Scientia 34B(4) (2014), 1287–1300.

  • [12]

    Ghahramani, H.: On derivations and Jordan derivations through zero products, Operator and Matrices 4 (2014), 759–771.

  • [13]

    Ghahramani, H.: Linear maps on group algebras determined by the action of the derivations or anti-derivations on a set of orthogonal elements, Results Math. 73 (2018), 132–146.

  • [14]

    Hou, J. C.—Zhang, X. L.: Ring isomorphisms and linear or additive maps preserving zero products on nest algebras, Linear Algebra Appl. 387 (2004), 343–360.

    • Crossref
    • Export Citation
  • [15]

    Jing, W. S.—Lu, S.—Li, P.: Characterizations of derivations on some operator algebras, Bull. Austr. Math. Soc. 66 (2002), 227–232.

    • Crossref
    • Export Citation
  • [16]

    Lu, F.: Characterizations of derivations and Jordan derivations on Banach algebras, Linear Algebra Appl. 430 (2009), 2233–2239.

    • Crossref
    • Export Citation
  • [17]

    Marcoux, L. W.: Projections, commutators and Lie ideals in C -algebras, Math. Proc. R. Ir. Acad. 110A (2010), 31–55

  • [18]

    Pearcy, C.—Topping, D.: Sum of small numbers of idempotent, Michigan Math. J. 14 (1967), 453–465.

    • Crossref
    • Export Citation
  • [19]

    Ponnusamy, S.—Silverman, H.: Complex Variables with Applications, Birkhäuser, Boston, 2006.

  • [20]

    Qi, X.—Hou, J.: Characterizations of derivations of Banach space nest algebras: All-derivable points, Linear Algebra Appl. 432 (2010), 3183–3200.

    • Crossref
    • Export Citation
  • [21]

    Sinclair, A. M.: Jordan homomorphisms and derivations on semisimple Banach algebras, Proc. Amer. Math. Soc. 24 (1970), 209–214.

  • [22]

    Zhu, J.—Xiong, C. P.: Generalized derivable mappings at zero point on some reflexive operator algebras, Linear Algebra Appl. 397 (2005), 367–379.

    • Crossref
    • Export Citation
  • [23]

    Zhu, J.—Xiong, C. P.: Derivable mappings at unit operator on nest algebras, Linear Algebra Appl. 422 (2007), 721–735.

    • Crossref
    • Export Citation
  • [24]

    Zhu, J.—Zhao, S.: Characterizations all-derivable points in nest algebras, Proc. Amer. Math. Soc. 141(7) (2013), 2343–2350.

    • Crossref
    • Export Citation
Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process.

Search