A short proof that B(L_1) is not amenable

Choi, Yemon (2021) A short proof that B(L_1) is not amenable. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 151 (6). pp. 1758-1767. ISSN 0308-2105

[thumbnail of 2009.04028v3]
Text (2009.04028v3)
2009.04028v3.pdf - Accepted Version
Available under License Creative Commons Attribution-NonCommercial-NoDerivs.

Download (159kB)


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.

Item Type:
Journal Article
Journal or Publication Title:
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Additional Information:
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.
Uncontrolled Keywords:
?? amenable banach algebrabanach spacesoperator idealsmathematics(all) ??
ID Code:
Deposited By:
Deposited On:
13 Oct 2020 09:55
Last Modified:
12 Feb 2024 00:38