2-Local derivations on matrix algebras and algebras of measurable operators (original) (raw)

Abstract

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 Mn(mathcalA)M_n(\mathcal{A})Mn(mathcalA) into Mn(mathcalM)M_n(\mathcal{M})Mn(mathcalM) (ngeq3)(n\geq 3)(ngeq3) 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.

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