B(l^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.

13 pages; final touchups

References in corpus (1)

• A hereditarily indecomposable L_\infty-space that solves the scalar-plus-compact problem