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

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.

Withdrawn due to a non-repairable flaw in the proof of Theorem 2.5