Characterizations of Lie centralizers of generalized matrix algebras. (English) Zbl 07834086
Summary: Let \(\mathcal{G}\) be a generalized matrix algebra. A linear map \(\phi : \mathcal{G} \to \mathcal{G}\) is said to be a left (right) Lie centralizer at \(E \in \mathcal{G}\) if \(\phi([S, T]) = [\phi(S), T] (\phi([S, T]) = [S, \phi(T)])\) holds for all \(S, T \in \mathcal{G}\) with \(ST = E\). \(\phi\) is of a standard form if \(\phi(A) = ZA + \gamma(A)\) for all \(A \in \mathcal{G}\), where \(Z\) is in the center of \(\mathcal{G}\) and \(\gamma\) is a linear map from \(\mathcal{G}\) into its center vanishing on each commutator \([S, T]\) whenever \(ST = E\). In this paper, we give a complete characterization of \(\phi\). It is shown that, under some suitable assumptions on \(\mathcal{G}\), \(\phi\) has a standard form.
MSC:
| 16W25 | Derivations, actions of Lie algebras |
| 47B47 | Commutators, derivations, elementary operators, etc. |
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