Can $ℬ(ℓ^{p})$ ever be amenable?

Matthew Daws ; Volker Runde

Studia Mathematica, Tome 187 (2008), p. 151-174 / Harvested from The Polish Digital Mathematics Library

Résumé

It is known that $ℬ(ℓ^{p})$ is not amenable for p = 1,2,∞, but whether or not $ℬ(ℓ^{p})$ is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if $ℬ(ℓ^{p})$ is amenable for p ∈ (1,∞), then so are $ℓ^{∞}(ℬ(ℓ^{p}))$ and $ℓ^{∞}((ℓ^{p}))$. Moreover, if $ℓ^{∞}((ℓ^{p}))$ is amenable so is $ℓ^{∞}(,(E))$ for any index set and for any infinite-dimensional $ℒ^{p}$-space E; in particular, if $ℓ^{∞}((ℓ^{p}))$ is amenable for p ∈ (1,∞), then so is $ℓ^{∞}((ℓ^{p} ⊕ ℓ²))$. We show that $ℓ^{∞}((ℓ^{p} ⊕ ℓ²))$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter over ℕ, we exhibit a closed left ideal of $((ℓ^{p}))_{}$ lacking a right approximate identity, but enjoying a certain very weak complementation property.

Détail

Publié le : 2008-01-01

Zbl 1145.47056

EUDML-ID : urn:eudml:doc:284734