2-Local derivations on matrix algebras over semi-prime Banach algebras and on $AW^\ast$-algebras

TL;DR

This paper proves that all 2-local derivations on matrix algebras over semi-prime Banach algebras and on $AW^\

Contribution

It establishes that 2-local derivations are actual derivations on these algebra classes, extending previous results to broader contexts.

Findings

2-local derivations on $M_{2^n}(\\mathcal{A})$ are derivations for semi-prime Banach algebras.

All 2-local derivations on $AW^\

The results unify the understanding of derivations in these algebraic structures.

Abstract

The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra $\mathcal{A}$ with the inner derivation property we prove that any 2-local derivation on the algebra $M_{2^n}(\mathcal{A}),$ $n\geq 2,$ is a derivation. We apply this result to $AW^\ast$-algebras and show that any 2-local derivation on an arbitrary $AW^\ast$-algebra is a derivation.