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2-local derivations on matrix algebras and algebras of measurable operators. (English) Zbl 1458.46055
Summary: Let \(\mathcal{A}\) be a unital Banach algebra such that any Jordan derivation from \(\mathcal{A}\) into any \(\mathcal{A}\)-bimodule \(\mathcal{M}\) is a derivation. We prove that any 2-local derivation from the algebra \(M_n(\mathcal{A})\) into \(M_n(\mathcal{M})\) (\(n\geq 3\)) is a derivation. We apply this result to show that any 2-local derivation on the algebra of locally measurable operators affiliated with a von Neumann algebra without direct abelian summands is a derivation.

MSC:
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
47B47 Commutators, derivations, elementary operators, etc.
47C15 Linear operators in \(C^*\)- or von Neumann algebras
16W25 Derivations, actions of Lie algebras
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