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It is known that ${\cal B}(\ell^p)$ is not amenable for $p =1,2,\infty$, butwhether or not ${\cal B}(\ell^p)$ is amenable for $p \in (1,\infty) \setminus\{2 \}$ is an open problem. We show that, if ${\cal B}(\ell^p)$ is amenable for$p \in (1,\infty)$, then so are $\ell^\infty({\cal B}(\ell^p))$ and$\ell^\infty({\cal K}(\ell^p))$. Moreover, if $\ell^\infty({\cal K}(\ell^p))$is amenable so is $\ell^\infty(\mathbb{I},{\cal K}(E))$ for any index set$\mathbb I$ and for any infinite-dimensional ${\cal L}^p$-space $E$; inparticular, if $\ell^\infty({\cal K}(\ell^p))$ is amenable for $p \in(1,\infty)$, then so is $\ell^\infty({\cal K}(\ell^p \oplus \ell^2))$. We showthat $\ell^\infty({\cal K}(\ell^p \oplus \ell^2))$ 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 $\cal U$ over $\posints$, we exhibit aclosed left ideal of $({\cal K}(\ell^p))_{\cal U}$ lacking a right approximateidentity, but enjoying a certain, very weak complementation property.

Can B(l^p) ever be amenable?