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

Abstract

Let $S(0,1)$ be the $\ast$-algebra of all classes of Lebesgue measurable functions on the unit interval $(0,1)$ and let $(\mathcal{A},{\Vert \cdot \Vert }_{\mathcal{A}})$ be a complete symmetric $\Delta$-normed $\ast$-subalgebra of $S(0,1)$, in which simple functions are dense, e.g., $L_{\infty }(0,1)$, $L_{\log }(0,1)$, $S(0,1)$ and the Arens algebra $L^{\omega }(0,1)$ equipped with their natural $\Delta$-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 $I_{\infty }$) 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 $\delta :\mathcal{A}\to S(\mathcal{A})$ can not be extended to a derivation on $S(\mathcal{M})$. In particular, we answer an untreated question in \cite{BKS1}.