Weak 2-local derivations on ${\mathbb{M}}_n$

Abstract

We introduce the notion of weak-2-local derivation (respectively, ${}^{\ast }$-derivation) on a C${}^{\ast }$-algebra $A$ as a (non-necessarily linear) map $\Delta :A\to A$ satisfying that for every $a,b\in A$ and $\varphi \in A^{\ast }$ there exists a derivation (respectively, a ${}^{\ast }$-derivation) $D_{a,b,\varphi }:A\to A$, depending on $a$, $b$ and $\varphi$, such that $\varphi \Delta (a)=\varphi D_{a,b,\varphi }(a)$ and $\varphi \Delta (b)=\varphi D_{a,b,\varphi }(b)$. We prove that every weak-2-local ${}^{\ast }$-derivation on $M_n$ is a linear derivation. We also show that the same conclusion remains true for weak-2-local ${}^{\ast }$-derivations on finite dimensional C${}^{\ast }$-algebras.