Characterizations of Jordan derivations on algebras of locally measurable operators

TL;DR

This paper investigates Jordan derivations on algebras of locally measurable operators affiliated with properly infinite von Neumann algebras, proving their continuity and extending derivations under certain conditions.

Contribution

It establishes the continuity of Jordan derivations on these algebras and provides methods to extend derivations from von Neumann algebras to their affiliated operator algebras.

Findings

Jordan derivations are continuous in the local measure topology

Extensions of Jordan derivations from von Neumann algebras to operator algebras are constructed

Continuity of Jordan derivations holds for subalgebras containing the von Neumann algebra

Abstract

We prove that if $\mathcal M$ is a properly infinite von Neumann algebra and $LS(\mathcal M)$ is the local measurable operator algebra affiliated with $\mathcal M$, then every Jordan derivation from $LS(\mathcal M)$ into itself is continuous with respect to the local measure topology $t(\mathcal M)$. We construct an extension of a Jordan derivation from $\mathcal M$ into $LS(\mathcal M)$ up to a Jordan derivation from $LS(\mathcal M)$ into itself. Moreover, we prove that if $\mathcal M$ is a properly von Neumann algebra and $\mathcal A$ is a subalgebra of $LS(\mathcal M)$ such that $\mathcal M\subset\mathcal A$, then every Jordan derivation from $\mathcal A$ into $LS(\mathcal M)$ is continuous with respect to the local measure topology $t(\mathcal M)$.