Abstract
Let \(\mathcal{A} \subseteq B(\mathcal{H})\) be a standard operator algebra on a Hilbert space \(\mathcal{H}\) where dim \(\mathcal{H} \geq 2\), and \(\mathcal{A}\) is closed under the adjoint operation. In this article, all linear maps \(\delta,\tau: \mathcal{A} \longrightarrow B(\mathcal{H})\) satisfying \(A\tau(B)^{*} +\delta(A)B^{*} = 0 (A^{*}\tau (B) + \delta(A)^{*}B = 0)\) whenever \(AB^{*} = 0 (A^{*}B = 0)\) are characterized, and as an application, linear maps on \(\mathcal{A}\) behaving like right (left) centralizers at orthogonal elements are described.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D.A. Abulhamil, F.B. Jamjoom and A.M. Peralta, Linear maps which are anti-derivable at zero, Bull. Malays. Math. Sci. Soc., 43 (2020), 4315–4334.
J. Alaminos, M. Brešar, J. Extremera and A.R. Villena, Maps preserving zero products, Studia Math., 193 (2009), 131–159.
A. Barari, B. Fadaee and H. Ghahramani, Linear maps on standard operator algebras characterized by action on zero products, Bull. Iran. Math. Soc., 45 (2019), 1573–1583.
D. Benkovič and M. Grašič, Generalized derivations on unital algebras determined by action on zero products, Linear Algebra Appl., 445 (2014), 347–368.
M. Brešar, Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinb. Sect. A., 137 (2007), 9–21.
M.Burgos and J.S.Ortega, On map**s preserving zero products, Linear Multilinear Algebra, 61 (2013), 323–335.
M.A.Chebotar, W.-F.Ke and P.-H. Lee, Maps characterized by action on zero products, Pacific J. Math, 216 (2004), 217–228.
H.G.Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, Oxford University Press, Oxford, 2000.
B. Fadaee and H.Ghahramani, Linear maps on \(C^*\)-algebras behaving like (anti-)derivations at orthogonal elements, Bull. Malays. Math. Sci. Soc., 43 (2020), 2851–2859.
B. Fadaee, K. Fallahi and H. Ghahramani, Characterization of linear map**s on (Banach)*-algebras by similar properties to derivations, Math. Slovaca, 70 (2020), 1003–1011.
A. Fošner and H.Ghahramani, Ternary derivations of nest algebras, Operator Matrices, 15 (2021), 327–339.
H. Ghahramani, Additive map**s derivable at nontrivial idempotents on Banach algebras, Linear Multilinear Algebra, 60 (2012), 725–742.
H.Ghahramani, On rings determined by zero products, J. Algebra Appl., 12 (2013), 1–15.
H.Ghahramani, Additive maps on some operator algebras behaving like \((\alpha,\beta)\)-derivations or generalized \((\alpha,\beta)\)-derivations at zero-product elements, Acta Math. Sci., 34B (2014), 1287–1300.
H.Ghahramani, On derivations and Jordan derivations through zero products, Operator Matrices, 8 (2014), 759–771.
H. Ghahramani, 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.
H.Ghahramani and Z. Pan, Linear maps on *-algebras acting on orthogonal elements like derivations or anti-derivations, Filomat, 32 (2018), 4543–4554.
W. **g, S. Lu and P. Li, Characterization of derivation on some operator algebras, Bull. Austr. Math. Soc., 66 (2002), 227–232.
T.-K. Lee, Generalized skew derivations characterized by acting on zero products, Pacific J. Math., 216 (2004), 293–301.
J. Li and Z. Pan, Annihilator preserving maps, multipliers, and derivations, Linear Algebra Appl., 432 (2010), 5–13.
J.Li and S.Li, Linear map**s characterized by action on zero products or unit products, Bull. Iran. Math. Soc., (2021), https://doi.org/10.1007/s41980-020-00499-y.
L. Liu, Characterization of centralizers on nest subalgebras of von Neumann algebras by local action, Linear Multilinear Algebra, 64 (2016), 383–392.
X.F.Qi, Characterization of centralizers on rings and operator algebras, Acta Math. Sin. Chin. Ser., 56 (2013), 459–468.
W.S.Xu, R.L.An and J.C.Hou, Equivalent characterization of centralizers on B(H), Acta Math. Sin. English Ser., 32 (2016), 1113–1120.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ghahramani, H., Mokhtari, A.H. Characterizing linear maps of standard operator algebras through orthogonality. Acta Sci. Math. (Szeged) 88, 777–786 (2022). https://doi.org/10.1007/s44146-022-00049-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s44146-022-00049-4