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) → B(H) satisfying the 3-local property is a *-monomorphism, that is, every linear mapping T : B(H) → 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) → B(H), depending on a, ξ, and η, such that T(a)(ξ) = π a,ξ,η (a)(ξ), and T(a)(η) = π a,ξ,η (a)(η), is a *-monomorphism. This solves a question posed by Molnár in (Arch Math 102:83–89 2014).