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

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.