Can B(l^p) ever be amenable?

Abstract

It is known that $\mathcal{B}({\ell }^p)$ is not amenable for $p=1,2,\infty$, but whether or not $\mathcal{B}({\ell }^p)$ is amenable for $p\in (1,\infty )\setminus \{2\}$ is an open problem. We show that, if $\mathcal{B}({\ell }^p)$ is amenable for $p\in (1,\infty )$, then so are ${\ell }^{\infty }(\mathcal{B}({\ell }^p))$ and ${\ell }^{\infty }(\mathcal{K}({\ell }^p))$. Moreover, if ${\ell }^{\infty }(\mathcal{K}({\ell }^p))$ is amenable so is ${\ell }^{\infty }(\mathbb{I},\mathcal{K}(E))$ for any index set $\mathbb{I}$ and for any infinite-dimensional ${\mathcal{L}}^p$-space $E$; in particular, if ${\ell }^{\infty }(\mathcal{K}({\ell }^p))$ is amenable for $p\in (1,\infty )$, then so is ${\ell }^{\infty }(\mathcal{K}({\ell }^p\oplus {\ell }^2))$. We show that ${\ell }^{\infty }(\mathcal{K}({\ell }^p\oplus {\ell }^2))$ is not amenable for $p=1,\infty$, but also that our methods fail us if $p\in (1,\infty )$. Finally, for $p\in (1,2)$ and a free ultrafilter $\mathcal{U}$ over $\posints$, we exhibit a closed left ideal of $(\mathcal{K}({\ell }^p){)}_{\mathcal{U}}$ lacking a right approximate identity, but enjoying a certain, very weak complementation property.

25 pages; cleaned up