Quasi-linear functionals determined by weak-2-local $^*$-derivations on $B(H)$

Abstract: We prove that, for every separable complex Hilbert space $H$, every weak-2-local $^*$-derivation on $B(H)$ is a linear $^*$-derivation. We also establish that every (non-necessarily linear nor continuous) weak-2-local derivation on a finite dimensional C$^*$-algebra is a linear derivation.