International Journal of Nonlinear Analysis and Applications

Characterization of multipliers on ‎${‎\star}‎$‎-algebras acting on orthogonal elements

Document Type : Research Paper

Authors

Department of Mathematics, Ayatollah Borujerdi University, Borujerd, Iran

Abstract

Let $A$ be a Banach ${\star}$-algebra, $X$ be a Banach ${\star}$-$A$-bimodule and $T: A\longrightarrow X$ be a continuous linear map. In this paper, by using orthogonality conditions on $A$, we characterize the map $T$ on certain Banach algebra including $C^{\star}$-algebras, group algebras, standard operator algebras and Banach algebras that is generated by idempotents. We also characterize a continuous linear map from zero Jordan product determined Banach algebra $A$ into a Banach $A$-bimodule $X$, and give some applications of this result. 

Keywords

[1] J. Alaminos, M. Bresar, J. Extremera and A.R. Villena, Maps preserving zero products, Studia Math. 193 (2009), 131–159.
[2] J. Alaminos, M. Bresar, J. Extremera and A.R. Villena, Characterizing Jordan maps on C∗-algebras through zero products, Proc. Roy. Soc. Edinb. 53 (2010), 543–555.
[3] J. Alaminos, M. Bresar, J. Extremera and A.R. Villena, Zero Jordan product determained Banach algebras, J. Aust. Math. Soc. 193 (2020), 1–14.
[4] G. An, J. Li and J. He, Zero Jordan product determined algebras, Linear Algebra Appl. 475 (2015), 90–93.
[5] M. Bresar, Multiplication algebra and maps determined by zero products, Linear Multilinear Algebras 60 (2012), no. 7, 763–768.
[6] M. Burgos and J. Sanchez-Ortega, On mappings preserving zero products, Linear Multilinear Algebras, 61 (2013), no. 3, 323–335.
[7] H.G. Dales, Banach Algebras and Automatic Continuity, LMS Monographs 24, Clarendon Press, Oxford, 2000.
[8] B. Fadaee and H. Ghahramani, Linear maps on C⋆-algebras behaving like (anti-)derivations at orthogonal elements, Bull. Malays. Math. Sci. Soc. 38 (2019), no. 1, 155–164.
[9] H. Farhadi, Characterizing left or right centralizers on ⋆-algebras through orthogonal elements, Math. Anal. Conv. Optim. 3 (2022), no. 1, 37–41.
[10] H. Ghahramani and Z. Pan, Linear maps on ⋆-algebras acting on orthogonal element like derivations or antiderivations, Filomat 13 (2018), 4543–4554.
[11] H. Ghahramani, Linear maps on group algebras determined by the action of the derivations or anti-derivations on a set of orthogonal element, Results Math. 73 (2018), 1–14.
[12] H. Ghahramani, Lie centralizers at zero products on a class of operator algebras, Ann. Funct. Anal. 12 (2021), 34.
[13] B. E. Johnson, An introduction to the theory of centralizers, Proc. London Math. Soc. 14 (1964), 299–320.
[14] R. Larsen, An Introduction to the Theory of Multipliers, Berlin, New York, Springer-Verlag, 1971.
[15] A. Zivari-Kazempour, Characterization of n-Jordan multipliers, Vietnam J. Math. 50 (2022), 87–94.
[16] A. Zivari-Kazempour, Characterizing n-multipliers on Banach algebras through zero products, Int. J. Nonlinear Anal. Appl. 14 (2023), no. 1, 1071–1078.
[17] A. Zivari-Kazempour and M. Valaei, Characterization of n-Jordan multipliers through zero products, J. Anal. 30 (2022), 1059–1067.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 5
May 2024
Pages 49-56
  • Receive Date: 18 February 2023
  • Accept Date: 06 June 2023