(Non-)amenability of ℬ(E)

Volker Runde

Banach Center Publications (2010)

  • Volume: 91, Issue: 1, page 339-351
  • ISSN: 0137-6934

Abstract

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In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that ℬ ( E ) = ( E ) + ℂ i d E . Still, ℬ(ℓ²) is not amenable, and in the past decade, ℬ ( ℓ p ) was found to be non-amenable for p = 1,2,∞ thanks to the work of C. J. Read, G. Pisier, and N. Ozawa. We survey those results, and then-based on joint work with M. Daws-outline a proof that establishes the non-amenability of ℬ ( ℓ p ) for all p ∈ [1,∞].

How to cite

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Volker Runde. "(Non-)amenability of ℬ(E)." Banach Center Publications 91.1 (2010): 339-351. <http://eudml.org/doc/286427>.

@article{VolkerRunde2010,
abstract = {In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that $ℬ(E) = (E) + ℂid_\{E\}$. Still, ℬ(ℓ²) is not amenable, and in the past decade, $ℬ(ℓ^\{p\})$ was found to be non-amenable for p = 1,2,∞ thanks to the work of C. J. Read, G. Pisier, and N. Ozawa. We survey those results, and then-based on joint work with M. Daws-outline a proof that establishes the non-amenability of $ℬ(ℓ^\{p\})$ for all p ∈ [1,∞].},
author = {Volker Runde},
journal = {Banach Center Publications},
keywords = {amenable Banach algebras; -spaces},
language = {eng},
number = {1},
pages = {339-351},
title = {(Non-)amenability of ℬ(E)},
url = {http://eudml.org/doc/286427},
volume = {91},
year = {2010},
}

TY - JOUR
AU - Volker Runde
TI - (Non-)amenability of ℬ(E)
JO - Banach Center Publications
PY - 2010
VL - 91
IS - 1
SP - 339
EP - 351
AB - In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that $ℬ(E) = (E) + ℂid_{E}$. Still, ℬ(ℓ²) is not amenable, and in the past decade, $ℬ(ℓ^{p})$ was found to be non-amenable for p = 1,2,∞ thanks to the work of C. J. Read, G. Pisier, and N. Ozawa. We survey those results, and then-based on joint work with M. Daws-outline a proof that establishes the non-amenability of $ℬ(ℓ^{p})$ for all p ∈ [1,∞].
LA - eng
KW - amenable Banach algebras; -spaces
UR - http://eudml.org/doc/286427
ER -

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