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 ${}^{\ast }$-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 ${}^{\ast }$-automorphism ${\pi }_{a,\xi ,\eta }:B(H)\to B(H)$, depending on $a$, $\xi$, and $\eta$, such that $$T(a)(\xi )={\pi }_{a,\xi ,\eta }(a)(\xi ),andT(a)(\eta )={\pi }_{a,\xi ,\eta }(a)(\eta ),$$ is a ${}^{\ast }$-monomorphism. This solves a question posed by L. Molnár in [\emph{Arch. Math.} \textbf{102}, 83-89 (2014)].