$\mathcal {B}(\ell ^p)$ is never amenable

Abstract:

We show that if $E$ is a Banach space with a basis satisfying a certain condition, then the Banach algebra $\ell ^\infty ({\mathcal K}(\ell ^2 \oplus E))$ is not amenable; in particular, this is true for $E = \ell ^p$ with $p \in (1,\infty )$. As a consequence, $\ell ^\infty ({\mathcal K}(E))$ is not amenable for any infinite-dimensional ${\mathcal L}^p$-space. This, in turn, entails the non-amenability of ${\mathcal B}(\ell ^p(E))$ for any ${\mathcal L}^p$-space $E$, so that, in particular, ${\mathcal B}(\ell ^p)$ and ${\mathcal B}(L^p[0,1])$ are not amenable.