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

Abstract

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 $\xi,\eta$ in $H$, there exists a $^*$-automorphism $\pi_{a,\xi,\eta}: B(H)\to B(H)$, depending on $a$, $\xi$, and $\eta$, such that $$T(a) (\xi) = \pi_{a,\xi,\eta} (a) (\xi), \hbox{ and } T(a) (\eta) = \pi_{a,\xi,\eta} (a) (\eta),$$ is a $^*$-monomorphism. This solves a question posed by L. Moln\'ar in [\emph{Arch. Math.} \textbf{102}, 83-89 (2014)].