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M-embedded symmetric operator spaces and the derivation problem

Part of: Selfadjoint operator algebras Linear function spaces and their duals

Published online by Cambridge University Press:  20 August 2019

JINGHAO HUANG ,
GALINA LEVITINA  and
FEDOR SUKOCHEV
JINGHAO HUANG
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, NSW, Australia. e-mails: jinghao.huang@unsw.edu.au, g.levitina@unsw.edu.au, f.sukochev@unsw.edu.au
GALINA LEVITINA
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, NSW, Australia. e-mails: jinghao.huang@unsw.edu.au, g.levitina@unsw.edu.au, f.sukochev@unsw.edu.au
FEDOR SUKOCHEV
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, NSW, Australia. e-mails: jinghao.huang@unsw.edu.au, g.levitina@unsw.edu.au, f.sukochev@unsw.edu.au
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Abstract

Let ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

MSC classification

Primary: 46L52: Noncommutative function spaces 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 3 , November 2020 , pp. 607 - 622
Copyright
© Cambridge Philosophical Society 2019

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