Can $ℬ(ℓ^{p})$ ever be amenable?

• Autorzy: Matthew Daws , Volker Runde
• Języki publikacji: EN

Abstrakty

EN

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 $ℓ^{∞}(𝓚(ℓ^{p}))$. Moreover, if $ℓ^{∞}(𝓚(ℓ^{p}))$ is amenable so is $ℓ^{∞}(𝕀,𝓚(E))$ for any index set 𝕀 and for any infinite-dimensional $ℒ^{p}$-space~E; in particular, if $ℓ^{∞}(𝓚(ℓ^{p}))$ is amenable for p ∈ (1,∞), then so is $ℓ^{∞}(𝓚(ℓ^{p} ⊕ ℓ²))$. We show that $ℓ^{∞}(𝓚(ℓ^{p} ⊕ ℓ²))$ 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 𝒰 over ℕ, we exhibit a closed left ideal of $(𝓚(ℓ^{p}))_{𝒰}$ lacking a right approximate identity, but enjoying a certain very weak complementation property.

Kategorie tematyczne

• 47L20: Operator ideals
• 47L10: Algebras of operators on Banach spaces and other topological linear spaces
• 46H20: Structure, classification of topological algebras
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 46B45: Banach sequence spaces
• 46B08: Ultraproduct techniques in Banach space theory
• 46B07: Local theory of Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2008
• Tom: 188
• Numer: 2
• Strony: 151-174

Daty

wydano

2008

Twórcy

autor

Matthew Daws

• Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

autor

Volker Runde

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

Identyfikatory

DOI

10.4064/sm188-2-4