On a generalized Semrl's theorem for weak-2-local derivations on B(H)

TL;DR

This paper proves that weak-2-local derivations on operator algebras like B(H), K(H), atomic von Neumann algebras, and compact C*-algebras are necessarily linear derivations, extending Semrl's theorem.

Contribution

It generalizes Semrl's theorem by showing weak-2-local derivations are linear on broader classes of operator algebras.

Findings

Weak-2-local derivations on B(H) are linear derivations.

Weak-2-local derivations on K(H) are linear derivations.

Weak-2-local derivations on atomic von Neumann algebras are linear.

Abstract

We prove that, for every complex Hilbert space $H$, every weak-2-local derivation on $B(H)$ or on $K(H)$ is a linear derivation. We also establish that every weak-2-local derivation on an atomic von Neumann algebra or on a compact C$^*$-algebra is a linear derivation.