(Non-)amenability of ℬ(E)

• Autorzy: Volker Runde
• Języki publikacji: EN

Abstrakty

EN

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 $ℬ(E) = 𝒦(E) + ℂid_{E}$. 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,∞].

Kategorie tematyczne

• 47L10: Algebras of operators on Banach spaces and other topological linear spaces
• 46H20: Structure, classification of topological algebras
• 46B45: Banach sequence spaces
• 46B07: Local theory of Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2010
• Tom: 91
• Numer: 1
• Strony: 339-351

Daty

wydano

2010

Twórcy

autor

Volker Runde

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

Identyfikatory

DOI

10.4064/bc91-0-20