Research Article

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

Published in: Quaestiones Mathematicae
Volume 46 , issue 5 , 2023 , pages: 909–926

DOI: 10.2989/16073606.2022.2053225

Author(s): Aleksey Ber National University of Uzbekistan, Uzbekistan , Jinghao Huang University of New South Wales, Australia , Karimbergen Kudaybergenov Karakalpak State University, Uzbekistan , Fedor Sukochev University of New South Wales, Australia

Keywords: 46L57 , 47B47 , 46L51 , 26A24 , Murray-von Neumann algebras , derivations , approximately differentiable functions , dyadic translations

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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 L^ω (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].

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