Can ${\cal B}(\ell^p)$ ever be amenable?

Matthew Daws ¹   ; Volker Runde ²

¹ Department of Pure Mathematics University of Leeds Leeds, LS2 9JT, United Kingdom
² Department of Mathematical and Statistical Sciences University of Alberta Edmonton, AB, Canada T6G 2G1

Studia Mathematica, Tome 188 (2008) no. 2, pp. 151-174

Résumé

It is known that ${\cal B}(\ell^p)$ is not amenable for $p =1,2,\infty$, but whether 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~$E$; in particular, 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 show that $\ell^\infty({\cal K}(\ell^p \oplus \ell^2))$ 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 $\mathbb N$, we exhibit a closed left ideal of $({\cal K}(\ell^p))_{\cal U}$ lacking a right approximate identity, but enjoying a certain very weak complementation property.

Détail

DOI : 10.4064/sm188-2-4

Keywords: known cal ell amenable infty whether cal ell amenable infty setminus problem cal ell amenable infty ell infty cal ell ell infty cal ell moreover ell infty cal ell amenable ell infty mathbb cal index set mathbb infinite dimensional cal p space particular ell infty cal ell amenable infty ell infty cal ell oplus ell ell infty cal ell oplus ell amenable infty methods fail infty finally ultrafilter cal mathbb exhibit closed ideal cal ell cal lacking right approximate identity enjoying certain weak complementation property