2-Local derivations on algebras of locally measurable operators

TL;DR

This paper proves that all 2-local derivations on the algebra of locally measurable operators affiliated with a type I$_ty$ von Neumann algebra are actually derivations, confirming a conjecture in operator algebra theory.

Contribution

It establishes that every 2-local derivation on $LS(M)$ for type I$_ty$ von Neumann algebras is a derivation, extending understanding of derivation structures.

Findings

All 2-local derivations on $LS(M)$ are derivations.

The result applies specifically to type I$_ty$ von Neumann algebras.

The proof confirms a conjecture in the theory of operator algebras.

Abstract

The paper is devoted to 2-local derivations on the algebra $LS(M)$ of all locally measurable operators affiliated with a type I$_\infty$ von Neumann algebra $M.$ We prove that every 2-local derivation on $LS(M)$ is a derivation.