(Non-)amenability of $\mathcal B(E)$ and Banach space geometry

TL;DR

This paper establishes broad geometric criteria for non-amenability of the algebra of bounded operators on various Banach spaces, significantly extending known results and providing shorter proofs for many classes of spaces.

Contribution

It introduces general geometric conditions that imply non-amenability of al B(E) for wide classes of Banach spaces, covering many new examples.

Findings

Most classical Banach spaces have non-amenable al B(E)

The criteria apply to al Lp, Lorentz, Orlicz, Schatten, James, Schlumprecht, and Tsirelson spaces

The only known amenable case is the Argyros--Haydon space

Abstract

Let $E$ be a Banach space, and $\mathcal B(E)$ the algebra of all bounded linear operators on $E$. The question of amenability of $\mathcal B(E)$ goes back to Johnson's seminal memoir \cite{johnson} from 1972. We present the first general criteria applying to very wide classes of Banach spaces, given in terms of the Banach space geometry of $E$, which imply that $\mathcal B(E)$ is non-amenable. We cover all spaces for which this is known so far (with the exception of one particular example), with much shorter proofs, such as $\ell_p$ for $p \in [1, \infty]$ and $c_0$, but also many new spaces: the numerous classes of spaces covered range from all $\mathcal{L}_p$-spaces for $p \in (1, \infty)$ to Lorentz sequence spaces and reflexive Orlicz sequence spaces, to the Schatten classes $S_p$ for $p \in [1,\infty]$, and to the James space $J$, the Schlumprecht space $S$, and the Tsirelson space $T$, among others. Our approach also highlights the geometric difference to the only space for which $\mathcal B(E)$ \emph{is} known to be amenable, the Argyros--Haydon space, which solved the famous scalar-plus-compact problem.