We prove that any derivation of the *-algebra $LS(\mathcal{ M})$ of all locally measurable operators affiliated with a properly infinite von Neumann algebra $\mathcal{ M}$ is continuous with respect to the local measure topology $t(\mathcal{ M})$. Building an extension of a derivation $\delta:\mathcal{ M}\longrightarrow LS(\mathcal{ M})$ up to a derivation from $LS(\mathcal{ M})$ into $LS(\mathcal{ M})$, it is further established that any derivation from $\mathcal{ M}$ into $LS(\mathcal{ M})$ is $t(\mathcal{ M})$-continuous.
Categories: math.OA