Characterizations of Jordan derivations on algebras of locally measurable operators

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 $t(\mathcal M)$.