Bilocal *-automorphisms of B(H) satisfying the 3-local property

We prove that, for a complex Hilbert space $H$ with dimension bigger or equal than three, every linear mapping $T: B(H)\to B(H)$ satisfying the 3-local property is a $^*$-monomorphism, that is, every linear mapping $T: B(H) \to B(H)$ satisfying that for every $a$ in $B(H)$ and every $ξ,η$ in $H$, there exists a $^*$-automorphism $π_{a,ξ,η}: B(H)\to B(H)$, depending on $a$, $ξ$, and $η$, such that $T(a) (ξ) = π_{a,ξ,η} (a) (ξ), \hbox{ and } T(a) (η) = π_{a,ξ,η} (a) (η),$ is a $^*$-monomorphism. This solves a question posed by L. Molnár in [\emph{Arch. Math.} \textbf{102}, 83-89 (2014)].

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