Weak-2-Local Derivations on $\mathbb M_n$

Mohsen Niazi, Antonio M. Peralta

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