A short proof that ℬ(L1) is not amenable. Issue 6 (December 2021) - Choi, Yemon
Abstract : Non-amenability of 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 ^∞ (K(_1)) and a transference principle developed by Daws and Runde (Studia Math., 2010). In this note, we provide a short proof that 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 B(L_1) is not even approximately amenable.