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

TL;DR

This paper proves that linear maps on B(H) satisfying a specific 3-local property are necessarily *-monomorphisms, resolving a question about local automorphisms in operator algebra theory.

Contribution

It establishes that 3-local property-preserving linear maps on B(H) are *-monomorphisms for Hilbert spaces of dimension at least three, confirming a conjecture by Molnár.

Findings

Linear maps satisfying the 3-local property are *-monomorphisms.

The result applies to Hilbert spaces with dimension ≥ 3.

It answers an open question in operator algebra theory.

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)].