Abstract
A linear mapping T on a JB\(^*\)-triple E is called triple derivable at orthogonal pairs if for every \(a,b,c\in E\) with \(a\perp b\) we have
We prove that for each bounded linear mapping T on a JB\(^*\)-algebra A the following assertions are equivalent:
- (a):
-
T is triple derivable at zero;
- (b):
-
T is triple derivable at orthogonal elements;
- (c):
-
There exists a Jordan \(^*\)-derivation \(D:A\rightarrow A^{**}\), a central element \(\xi \in A^{**}_{sa},\) and an anti-symmetric element \(\eta \) in the multiplier algebra of A, such that
$$\begin{aligned} T(a) = D(a) + \xi \circ a + \eta \circ a, \hbox { for all } a\in A; \end{aligned}$$ - (d):
-
There exist a triple derivation \(\delta : A\rightarrow A^{**}\) and a symmetric element S in the centroid of \(A^{**}\) such that \(T= \delta +S\).
The result is new even in the case of C\(^*\)-algebras. We next establish a new characterization of those linear maps on a JBW\(^*\)-triple which are triple derivations in terms of a good local behavior on Peirce 2-subspaces. We also prove that assuming some extra conditions on a JBW\(^*\)-triple M, the following statements are equivalent for each bounded linear mapping T on M:
- (a):
-
T is triple derivable at orthogonal pairs;
- (b):
-
There exists a triple derivation \(\delta : M\rightarrow M\) and an operator S in the centroid of M such that \(T = \delta + S\).
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Acknowledgements
First author supported by the Higher Education and Scientific Research Ministry in Tunisia, UR11ES52: Analyse, Géométrie et Applications. Second author partially supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European Regional Development Fund project no. PGC2018-093332-B-I00, Programa Operativo FEDER 2014-2020 and Consejería de Economía y Conocimiento de la Junta de Andalucía grant number A-FQM-242-UGR18, and Junta de Andalucía grant FQM375. The authors would like to express their gratitude to the anonymous reviewers for their valuable comments and suggestions.
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Essaleh, A.B.A., Peralta, A.M. A linear preserver problem on maps which are triple derivable at orthogonal pairs. RACSAM 115, 146 (2021). https://doi.org/10.1007/s13398-021-01082-8
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DOI: https://doi.org/10.1007/s13398-021-01082-8
Keywords
- C\(^*\)-algebra
- JB\(^*\)-algebra
- JB\(^*\)-triple
- derivations
- Triple derivations
- Generalized derivations
- Maps triple derivable at zero
- Maps triple derivable at orthogonal elements