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

Matthew Daws, and Volker Runde. "Can $ℬ(ℓ^{p})$ ever be amenable?." Studia Mathematica 188.2 (2008): 151-174. <http://eudml.org/doc/284734>.

@article{MatthewDaws2008,
abstract = {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.},
author = {Matthew Daws, Volker Runde},
journal = {Studia Mathematica},
keywords = {amenability; -spaces; maximal operator ideals; ultra-amenability},
language = {eng},
number = {2},
pages = {151-174},
title = {Can $ℬ(ℓ^\{p\})$ ever be amenable?},
url = {http://eudml.org/doc/284734},
volume = {188},
year = {2008},
}

TY - JOUR
AU - Matthew Daws
AU - Volker Runde
TI - Can $ℬ(ℓ^{p})$ ever be amenable?
JO - Studia Mathematica
PY - 2008
VL - 188
IS - 2
SP - 151
EP - 174
AB - 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.
LA - eng
KW - amenability; -spaces; maximal operator ideals; ultra-amenability
UR - http://eudml.org/doc/284734
ER -