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Local triple derivations on \(\mathrm{C}^\ast\)-algebras and \(\mathrm{JB}^\ast\)-triples. (English) Zbl 1312.47050

A triple derivation on a Jordan triple \(E\) is a linear mapping \(\delta: E\to E\) such that, for all \(a,b,c\in E\), \[ \delta(\{a, b, c\})=\{\delta(a), b, c\}+\{a, \delta(b), c\}+\{a, b, \delta(c)\}, \] where \(\{\cdot, \cdot, \cdot\}\) is the triple product on \(E\). A linear mapping \(T: E\to E\) is said to be a local triple derivation if, for all \(a\in E\), there exists a derivation \(\delta_a\) on \(E\) such that \(T(a)=\delta_a(a)\).
It has been shown that continuous local triple derivations on \(JBW^*\)-triples are triple derivations [M. Mackey, ibid. 45, No. 4, 811–824 (2013; Zbl 1294.17025)], but whether this could be extended to JB\(^*\)-triples remained an open problem. One of the achievements of the present paper is to answer this question positively. It is also shown that, in fact, every local triple derivation on a JB\(^*\)-triple is automatically continuous and is, therefore, a triple derivation. This immediately applies to \(C^*\)-algebras, of course. These algebras are precisely the setting of the final section of the paper, where the interplay existing between triple derivations and generalized (Jordan) derivations on \(C^*\)-algebras is revealed.

MSC:

47B47 Commutators, derivations, elementary operators, etc.
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
17C65 Jordan structures on Banach spaces and algebras
47C15 Linear operators in \(C^*\)- or von Neumann algebras
46L05 General theory of \(C^*\)-algebras
46L08 \(C^*\)-modules

Citations:

Zbl 1294.17025
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References:

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