ℬ(ℓ^{𝓅}) is never amenable

Runde, Volker ¹

¹ Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Journal of the American Mathematical Society, Tome 23 (2010) no. 4, pp. 1175-1185

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-10-00668-5

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