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

Abstract

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