2-Local derivations on matrix algebras over semi-prime Banach algebras and on $A{W}^{\ast }$-algebras

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 $M_{2^n}(\mathcal{A}),$ $n\ge 2,$ is a derivation. We apply this result to $A{W}^{\ast }$-algebras and show that any 2-local derivation on an arbitrary $A{W}^{\ast }$-algebra is a derivation.