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  • 2009.04028v3

    Rights statement: https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/short-proof-that-l1-is-not-amenable/5EB9501F273D24ABD38EDEAB8B9433AD The final, definitive version of this article has been published in the Journal, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 151 (6), pp 1758-1767 2021, © 2020 Cambridge University Press.

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    Available under license: CC BY-NC-ND: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

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A short proof that B(L_1) is not amenable

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published
<mark>Journal publication date</mark>31/12/2021
<mark>Journal</mark>Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Issue number6
Volume151
Number of pages10
Pages (from-to)1758-1767
Publication StatusPublished
Early online date6/11/20
<mark>Original language</mark>English

Abstract

Non-amenability of ${\mathcal B}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for $E= \ell_p$ and $E=L_p$ for all $1\leq p<\infty$. However, the arguments are rather indirect: the proof for $L_1$ 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 $L_1$, and shows that ${\mathcal B}(L_1)$ is not even approximately amenable.

Bibliographic note

https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/short-proof-that-l1-is-not-amenable/5EB9501F273D24ABD38EDEAB8B9433AD The final, definitive version of this article has been published in the Journal, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 151 (6), pp 1758-1767 2021, © 2020 Cambridge University Press.

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