2-local triple derivations on von Neumann algebras

TL;DR

This paper proves that every 2-local triple derivation on a von Neumann algebra, regardless of linearity or continuity, is actually a triple derivation, establishing a strong algebraic property.

Contribution

It demonstrates that 2-local triple derivations on von Neumann algebras are always genuine triple derivations, revealing a new algebraic reflexivity property.

Findings

All 2-local triple derivations are triple derivations.

The set of triple derivations is algebraically 2-reflexive.

The result holds without assumptions of linearity or continuity.

Abstract

We prove that every {\rm(}not necessarily linear nor continuous{\rm)} 2-local triple derivation on a von Neumann algebra $M$ is a triple derivation, equivalently, the set Der$_{t} (M)$, of all triple derivations on $M,$ is algebraically 2-reflexive in the set $\mathcal{M}(M)= M^M$ of all mappings from $M$ into $M$.