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

We introduce the notion of weak-2-local derivation (respectively, $^*$-derivation) on a C$^*$-algebra $A$ as a (non-necessarily linear) map $Δ: A\to A$ satisfying that for every $a,b\in A$ and $ϕ\in A^*$ there exists a derivation (respectively, a $^*$-derivation) $D_{a,b,ϕ}: A\to A$, depending on $a$, $b$ and $ϕ$, such that $ϕΔ(a) = ϕD_{a,b,ϕ} (a)$ and $ϕΔ(b) = ϕD_{a,b,ϕ} (b)$. 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.

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