Linear maps on Banach algebras which are Lie centralizers at a point
DOI:
https://doi.org/10.31489/2026m3/210-221Keywords:
centralizer, Lie centralizer, Lie triple centralizer, commuting map, C∗-algebra, von Neumann algebras, standard operator algebras, matrix algebrasAbstract
In this paper, we study continuous linear maps that behave as Lie centralizers at specific points, including identity products, zero products, and idempotent products. Our aim is to determine when local Lie centralizer conditions force a linear map to have a global centralizer-like structure. We first show that every continuous linear map satisfying the Lie centralizer condition at identity products is necessarily a commuting map. We then obtain explicit descriptions of such map on C∗-algebras, von Neumann algebras, standard operator algebras, and matrix algebras. For maps acting at zero products, we prove that under suitable assumptions they are Lie centralizers, and we derive corresponding structural characterizations. We also investigate Lie centralizers at idempotent products and establish several characterization theorems for von Neumann algebras and standard operator algebras. These results extend a number of earlier theorems on commuting maps, centralizers, and Lie centralizers, and provide new criteria under which local conditions imply global behavior. Moreover, our approach highlights the strong relationship between local algebraic conditions and the global structure of linear maps. Consequently, this paper contributes to the theory of linear preserver problems and centralizer-type mappings on Banach and operator algebras.
References
Beidar, K.I., Martindale, W.S., & Mikhalev, A.V. (1996). Rings with Generalized Identities. New York: Marcel Dekker Inc.
Breˇsar, M. (1992). On a generalization of the notion of centralizing mappings. Proceedings of the American Mathematical Society, 114(3), 641–649. https://doi.org/10.2307/2159383
Ghahramani, H., & Jing, W. (2021). Lie centralizers at zero products on a class of operator algebras. Annals of Functional Analysis, 12, Article 34. https://doi.org/10.1007/s43034-021-00123-y
Breˇsar, M. (1993). Centralizing mappings and derivations in prime rings. Journal of Algebra, 156(2), 385–394. https://doi.org/10.1006/jabr.1993.1080
Breˇsar, M. (2004). Commuting maps: a survey. Taiwanese Journal of Mathematics, 8(3), 361– 397. https://doi.org/10.11650/twjm/1500407660
Breˇsar, M. (2007). Characterizing homomorphisms, derivations and multipliers in rings with idempotents. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 137(1), 9–21. https://doi.org/10.1017/S0308210504001088
Zivari-Kazempour, A. (2023). Characterizing n-multipliers on Banach algebras through zero products. International Journal of Nonlinear Analysis and Applications, 14(1), 1071–1078. https://doi.org/10.22075/ijnaa.2022.25729.3111
Zivari-Kazempour, A. (2022). Characterization of n-Jordan multipliers. Vietnam Journal of Mathematics, 50, 87–94. https://doi.org/10.1007/s10013-021-00479-7
Foˇsner, A., & Jing, W. (2019). Lie centralizers on triangular rings and nest algebras. Advances in Operator Theory, 4(2), 342–350. https://doi.org/10.15352/aot.1804-1341
Ghahramani, H. (2024). Centralizer-like additive maps on the Lie structure of Banach algebras. Bulletin of the Malaysian Mathematical Sciences Society, 47, Article 37. https://doi.org/10.1007/s40840-023-01634-8
Burgos, M., & Sanchez-Ortega, J. (2013). On mappings preserving zero products. Linear and Multilinear Algebra, 61(3), 323–335. https://doi.org/10.1080/03081087.2012.678344
Ghahramani, H., Mokhtari, A.H., & Wei, F. (2024). Lie centralizers and commutant preserving maps on generalized matrix algebras. Journal of Algebra and Its Applications, 23(05). https://doi.org/10.1142/S0219498824501068
Breˇsar, M. (1991). Centralizing mappings on von Neumann algebras. Proceedings of the American Mathematical Society, 111(2), 501–510. https://doi.org/10.1090/S0002-9939-1991-1028283-2
Dales, H.G. (2000). Banach Algebras and Automatic Continuity. Oxford: Clarendon Press.
Fadaee, B., Ghahramani, H., & Jing, W. (2023). Lie triple centralizers on generalized matrix algebras. Quaestiones Mathematicae, 46(2), 281–300. https://doi.org/10.2989/16073606.2021.2013972
Alaminos, J., Breˇsar, M., Extremera, J., & Villena, A.R. (2009). Maps preserving zero products. Studia Mathematica, 193, 131–159. https://doi.org/10.4064/sm193-2-3
Breˇsar, M., Grasic, M., & Ortega, J.S. (2009). Zero product determined matrix algebras. Linear Algebra and its Applications, 430(5-6), 1486–1498. https://doi.org/10.1016/j.laa.2007.11.018
Zivari-Kazempour, A., Ghahramani, H., & Jing, W. (2024). Derivable maps at commutative products on Banach algebras. Acta Scientiarum Mathematicarum, 90, 165–174. https://doi.org/10.1007/s44146-023-00104-8
