B(l^p) is never amenable

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