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

TL;DR

This paper presents a concise proof demonstrating that the algebra of bounded operators on L_1 is not amenable, simplifying previous complex arguments and establishing non-amenability through classical properties of representable operators.

Contribution

It offers a new, streamlined proof of non-amenability of B(L_1), avoiding indirect methods used in prior approaches.

Findings

B(L_1) is not amenable.

B(L_1) is not approximately amenable.

The proof bypasses complex machinery of previous proofs.

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.