Characterizations of derivations and Jordan derivations on Banach algebras. (English) Zbl 1171.47030
Let \(\mathcal A\) be a Banach algebra and \(\mathcal M\) be an \(\mathcal A\)-bimodule. An element \(W\) in \(\mathcal A\) is called a left (resp., right) separating point of \(\mathcal M\) if \(WM=0\) (resp., \(MW=0\)) for \(M\in\mathcal M\) implies that \(M=0\). It is shown in the paper that if \(W\) is a left or right separating point of \(\mathcal M\) and \(\delta:{\mathcal M}\to{\mathcal M}\) is a continuous linear map, then the following are equivalent: (1) \(\delta(AB)=\delta(A)B+A\delta(B)\) for all \(A,B\in{\mathcal A}\) with \(AB=W\); (2) \(\delta\) is a Jordan derivation and satisfies \(\delta(WA)=\delta(W)A+W\delta(A)\) and \(\delta(AW)=\delta(A)W+A\delta(W)\) for all \(A\in\mathcal A\).
As another result, it is proved that, if \(P\) is a nontrivial idempotent element in \(\mathcal A\) which is faithful with respect to \(\mathcal M\) and \(\delta:{\mathcal M}\to{\mathcal M}\) is a linear map which satisfies \(\delta(AB)=\delta(A)B+A\delta(B)\) whenever \(AB=P\) for \(A,B\in\mathcal A\), then \(\delta\) is a derivation.
As another result, it is proved that, if \(P\) is a nontrivial idempotent element in \(\mathcal A\) which is faithful with respect to \(\mathcal M\) and \(\delta:{\mathcal M}\to{\mathcal M}\) is a linear map which satisfies \(\delta(AB)=\delta(A)B+A\delta(B)\) whenever \(AB=P\) for \(A,B\in\mathcal A\), then \(\delta\) is a derivation.
Reviewer: Madjid Mirzavaziri (Mashhad)
MSC:
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 47B48 | Linear operators on Banach algebras |
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