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

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.