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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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