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A short proof that ℬ(L1) is not amenable

Part of: Linear spaces and algebras of operators Normed linear spaces and Banach spaces; Banach lattices Topological algebras, normed rings and algebras, Banach algebras Measures, integration, derivative, holomorphy

Published online by Cambridge University Press:  06 November 2020

Yemon Choi Open the ORCID record for Yemon Choi [Opens in a new window]
Yemon Choi*
Affiliation:
Department of Mathematics and Statistics, Lancaster University, LancasterLA1 4YF, UK (y.choi1@lancaster.ac.uk)
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Abstract

Non-amenability of ${\mathcal {B}}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E = ℓp and E = Lp for all 1 ⩽ p < ∞. However, the arguments are rather indirect: the proof for L1 goes via non-amenability of $\ell ^\infty ({\mathcal {K}}(\ell _1))$ and a transference principle developed by Daws and Runde (Studia Math., 2010).

In this note, we provide a short proof that ${\mathcal {B}}(L_1)$ and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on L1, and shows that ${\mathcal {B}}(L_1)$ is not even approximately amenable.

Keywords

Amenable Banach algebrasBanach spacesoperator idealsrepresentable operators

MSC classification

Primary: 46H10: Ideals and subalgebras 47L10: Algebras of operators on Banach spaces and other topological linear spaces
Secondary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties 46G10: Vector-valued measures and integration
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 1758 - 1767
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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