Weak-2-local derivations on Mn

Abstract

We introduce the notion of weak-2-local derivation (respectively, *-derivation) on a C*-algebra A as a (non-necessarily linear) map ∆ : A → A satisfying that for every a, b ∈ A and φ ∈ A* there exists a derivation (respectively, a *-derivation) Da,b,φ : A → A, depending on a, b and φ, such that φ∆(a) = φDa,b,φ(a) and φ∆(b) = φDa,b,φ(b). We prove that every weak-2-local *-derivation on Mn is a linear derivation. We also show that the same conclusion remains true for weak-2-local *-derivations on finite dimensional C*-algebras.