A short proof that ℬ(L1) is not amenable. Issue 6 (December 2021) - Choi, Yemon
Abstract : Non-amenability of ${\mathcal {B}}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E = ℓ p and E = L p for all 1 ⩽ p < ∞. However, the arguments are rather indirect: the proof for L 1 goes via non-amenability of $\ell ^∞ ({\mathcal {K}}(\ell _1))$ 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.