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 (A, || ||_A) be a complete symmetric ∆-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 ∆-norms. We show that there exists no non-trivial derivation _ : A→ S(0, 1) commuting with all dyadic translations of the unit interval. Let M be a type II (or I_ꭓ) von Neumann algebra, A be an arbitrary abelian von Neumann subalgebra of M, let S(M) be the algebra of all measurable operators affiliated with M. We show that there exists no non-trivial derivation _ : A→ S(A) which admits an extension to a derivation on S(M). In particular, we answer an untreated question in [8].