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A short proof that B(L1) is not amenable


Yemon Choi

* Corresponding author


Keywords: amenable Banach algebras, Banach spaces, operator ideals, representable operators.

MSC 2020: 46H10, 47L10 (primary); 46B22, 46G10 (secondary)

DOI: 10.1017/prm.2020.79


To appear in Proc. Roy. Soc. Edinburgh Sect. A (10 pages). Published online November 2020.

Preprint version available at arXiv 2009.04028



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 $1\leq p \lt \infty$. 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.


Quoting from the acknowledgements section of this paper:

The question of whether the ideal ${\mathfrak R}$ has either a left or right bounded approximate identity emerged from discussions with Matt Daws and Jon Bannon about a completely different problem in von Neumann algebras.

On that theme: it's good to remember from time to time that mathematics can still occasionally be about exploration and not just about fracking.

Yemon Choi