Non-existence of translation-invariant derivations on algebras of measurable functions

Let $S(0,1)$ be the $\ast$-algebra of all classes of Lebesgue measurable functions on the unit interval $(0,1)$ and let $($\mathcal{A}$,\left|\cdot \right|_$\mathcal{A}$)$beacompletesymmetric$\Delta$-normed$*$-subalgebra of $S(0,1)$, in which simple functions are dense, e.g., $L_{\mathrm{\infty }}(0,1)$, $L_{\log }(0,1)$, $S(0,1)$ and the Arens algebra $L^{\omega }(0,1)$ equipped with their natural $\mathrm{\Delta }$-norms. We show that there exists no non-trivial derivation $\delta :\mathcal{A}\to S(0,1)$ commuting with all dyadic translations of the unit interval. Let $\mathcal{M}$ be a type $II$ (or $I_{\mathrm{\infty }}$) von Neumann algebra, $\mathcal{A}$ be its abelian von Neumann subalgebra, let $S(\mathcal{M})$ be the algebra of all measurable operators affiliated with $\mathcal{M}$. We show that any non-trivial derivation $\delta:$\mathcal{A}$ \to S($\mathcal{A}$)$cannotbeextendedtoaderivationon$S($\mathcal{M}$)$. In particular, we answer an untreated question in \cite{BKS1}.