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

Let S(0,1) be the *-algebra of all classes of Lebesgue measurable functions on the unit interval (0,1) and let (Figure presented.) be a complete symmetric Δ-normed *-subalgebra of S(0,1), in which simple functions are dense, e.g., L _∞(0,1), L _log(0,1), S(0,1) and the Arens algebra L^ω (0,1) equipped with their natural Δ-norms. We show that there exists no non-trivial derivation (Figure presented.) commuting with all dyadic translations of the unit interval. Let (Figure presented.) be a type II (or I _∞) von Neumann algebra, (Figure presented.) be an arbitrary abelian von Neumann subalgebra of (Figure presented.), let (Figure presented.) be the algebra of all measurable operators affiliated with (Figure presented.). We show that there exists no non-trivial derivation (Figure presented.) which admits an extension to a derivation on (Figure presented.). In particular, we answer an untreated question in [8].