Multiplicative \(\ast\)-Lie triple higher derivations of standard operator algebras. (English) Zbl 07111860
Summary: Let \(\mathfrak{A}\) be a standard operator algebra on an infinite dimensional complex Hilbert space \(\mathcal{H}\) containing identity operator \(I\). In this paper it is shown that if \(\mathfrak{A}\) is closed under the adjoint operation, then every multiplicative \(\ast\)-Lie triple derivation \(d:\mathfrak{A}\to \mathcal{B}(\mathcal{H})\) is a linear \(\ast\)-derivation. Moreover, if there exists an operator \(S \in \mathcal{B}(\mathcal{H})\) such that \(S + S^\ast = 0\) then \(d(U)=US-SU\) for all \(U \in \mathfrak{A}\), that is, \(d\) is inner. Furthermore, it is also shown that any multiplicative \(\ast\)-Lie triple higher derivation \(D=\{\delta_n\}_{n\in\mathbb{N}}\) of \(\mathfrak{A}\) is automatically a linear inner higher derivation on \(\mathfrak{A}\) with \(d(U)^\ast = d(U^\ast)\).
MSC:
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 16W25 | Derivations, actions of Lie algebras |
| 46K15 | Hilbert algebras |
Keywords:
multiplicative \(\ast\)-Lie derivation; multiplicative \(\ast\)-Lie triple higher derivation; standard operator algebraReferences:
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