Characterizations of Jordan derivations on algebras of locally measurable operators

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 $t(\mathcal{M})$.