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

## A short proof that B(L1) is not amenable

Non-amenability of B(E) has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E= lp and E=Lp for all 1 ≤ p < ∞. However, the arguments are rather indirect: the proof for L1 goes via non-amenability of l(K(l1)) and a transference principle developed by Daws and Runde (Studia Math., 2010). In this note, we provide a short proof that B(L1) 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 L1, and shows that B(L1) is not even approximately amenable.

Update 2020-11-06: To appear in Proceedings of the Royal Society of Edinburgh, Section A. Published online as DOI: 10.1017/prm.2020.79

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1. 10 September, 2020 3:30 pm

$\mathfrak {It's~all~just~fraktured~faery~tales}~\ldots$

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