Local derivations on Rings containing a von Neumann algebra and a question of Kadison

TL;DR

This paper proves that for von Neumann algebras with a discrete abelian summand, all local derivations on affiliated measurable operators are actual derivations, resolving a question posed by Kadison.

Contribution

It establishes that local derivations on certain operator algebras are always derivations, answering an open question of Kadison.

Findings

All local derivations on the algebra of measurable operators affiliated with M are derivations.

The result applies to von Neumann algebras with a discrete abelian summand.

It confirms a conjecture posed by Richard Kadison.

Abstract

We prove that if M is a von Neumann algebra whose abelian summand is discrete, then every local derivation on the algebra of all measurable operators affilated with M is a derivation. This answers a question of Richard Kadison.