A short proof that $\mathcal{B}(L_1)$ is not amenable

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 $1\le p<\infty$. 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.

v3: AMS-LaTeX, 10 pages. Incorporates referee's feedback; typos fixed and one reference added. Final version, to appear in Proc. Roy. Soc. Edinburgh Sect. A