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Amenability and semisimplicity for second duals of quotients of the Fourier algebra A(G)

Part of: Topological algebras, normed rings and algebras, Banach algebras Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Edmond E. Granirer
Edmond E. Granirer
Affiliation:
Department of Mathematics University of British ColumbiaVancouver V6T 1Z2Canada e-mail: granirer@math.ubc.ca
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Abstract

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Let F ⊂ G be closed and A(F) = A(G)/IF. If F is a Helson set then A(F)** is an amenable (semisimple) Banach algebra. Our main result implies the following theorem: Let G be a locally compact group, F ⊂ G closed, a ∈ G. Assume either (a) For some non-discrete closed subgroup H, the interior of F ∩ aH in aH is non-empty, or (b) R ⊂ G, S ⊂ R is a symmetric set and aS ⊂ F. Then A(F)** is a non-amenable non-semisimple Banach algebra. This raises the question: How ‘thin’ can F be for A(F)** to remain a non-amenable Banach algebra?

MSC classification

Secondary: 46H20: Structure, classification of topological algebras 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A20: $L^1$-algebras on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 63 , Issue 3 , December 1997 , pp. 289 - 296
Copyright
Copyright © Australian Mathematical Society 1997

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