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

Volume 188 / 2008

Matthew Daws, Volker Runde Studia Mathematica 188 (2008), 151-174 MSC: Primary 47L10; Secondary 46B07, 46B08, 46B45, 46E30, 46H20, 47L20. DOI: 10.4064/sm188-2-4

Abstract

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.

Authors

  • Matthew DawsDepartment of Pure Mathematics
    University of Leeds
    Leeds, LS2 9JT, United Kingdom
    e-mail
  • Volker RundeDepartment of Mathematical
    and Statistical Sciences
    University of Alberta
    Edmonton, AB, Canada T6G 2G1
    e-mail

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