# 2-Local derivations on matrix algebras and algebras of measurable operators

## TL;DR

This paper proves that 2-local derivations on matrix algebras over certain Banach algebras and on algebras of measurable operators affiliated with von Neumann algebras are actual derivations, extending known results.

## Contribution

It establishes that 2-local derivations are derivations on matrix algebras over Banach algebras satisfying specific conditions and on algebras of measurable operators.

## Key findings

- 2-local derivations on M_n(A) are derivations for na3 3
- 2-local derivations on algebras of measurable operators are derivations
- Results apply to von Neumann algebras without abelian summands

## 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 $M_n(\mathcal{A})$ 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.