Local Derivations on Algebras of Measurable Operators

TL;DR

This paper investigates local derivations on algebras of measurable operators associated with von Neumann algebras, proving their equivalence to derivations under certain conditions and characterizing cases where non-derivation local derivations exist.

Contribution

It establishes that continuous local derivations on these algebras are derivations and characterizes when non-derivation local derivations occur, especially in type I von Neumann algebras.

Findings

Every continuous local derivation is a derivation.

All local derivations are inner in type I von Neumann algebras.

Conditions for existence of non-derivation local derivations in commutative cases.

Abstract

The paper is devoted to local derivations on the algebra $S(\mathcal{M},\tau)$ of $\tau$-measurable operators affiliated with a von Neumann algebra $\mathcal{M}$ and a faithful normal semi-finite trace $\tau.$ We prove that every local derivation on $S(\mathcal{M},\tau)$ which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the condition of continuity of the local derivation is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra $\mathcal{M}$ for the existence of local derivations which are not derivations on algebras of measurable operators affiliated with $\mathcal{M}.$