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

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

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: [email protected]
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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

References

[BM]Brown, G. and Moran, W., ‘Point derivations in M(G)’, Bull. London Math. Soc. 8 (1976), 57–64.CrossRefGoogle Scholar
[CL]Curtis, P. C. Jr and Loy, R. J., ‘The structure of amenable Banach algebras’, J. London Math. Soc. 40 (1989), 89–104.CrossRefGoogle Scholar
[DH]Duncan, J. and Hosseinium, S. A. R., ‘The second dual of a Banach algebra’, Proc. Royal Soc. Edinburgh Sect. A 84 (1979), 309–325.CrossRefGoogle Scholar
[Ey]Eymard, P., ‘L'algèbre de Fourier d'un groupe localement compact’, Bull. Soc. Math. France 92 (1964), 181–236.CrossRefGoogle Scholar
[Fo1]Forrest, B., ‘Amenability and the structure of the algebras Ap(G)’, Trans. Amer. Math. Soc. 343 (1994), 233–243.Google Scholar
[Fo2]Forrest, B., ‘Amenability and weak amenability of the Fourier algebra’, preprint.Google Scholar
[Go]Gourdeau, F., Amenability of Banach algebras (Ph. D. Thesis, Cambridge University, 1989).CrossRefGoogle Scholar
[Gr1]Granirer, E., ‘On some spaces of linear functionals on the algebras Ap(G) for locally compact groups’, Colloq. Math. 52 (1987), 119–132.CrossRefGoogle Scholar
[Gr2]Granirer, E., ‘On convolution operators with small support which are far from being convolution by a bounded measure’, Colloq. Math. 67 (1994), 33–60.CrossRefGoogle Scholar
[Gr3]Granirer, E., ‘Day points for quotients of the Fourier algebra A(G), extreme nonergodicity of their duals and extreme non Arens regularity’, Illinois J. Math. 40 (1996), 402–419.CrossRefGoogle Scholar
[Gr4]Granirer, E., ‘On the set of topologically invariant means on an algebra of convolution operators on L p(G)’, Proc. Amer. Math. Soc. 124 (1996), 3399–3406.CrossRefGoogle Scholar
[GLW]Ghahramani, F., Loy, R. J. and Willis, G. A., ‘Amenability and weak amenability of second conjugate Banach algebras’, Proc. Amer. Math. Soc. 124 (1996), 1489–1497.CrossRefGoogle Scholar
[GMc]Graham, C. C. and McGehee, O. C., Essays in commutative harmonic analysis (Springer, New York, 1979).CrossRefGoogle Scholar
[Hz1]Herz, C., ‘Harmonic synthesis for subgroups’, Ann. Inst. Fourier (Grenoble) 37 (1973), 91–123.CrossRefGoogle Scholar
[Hz2]Herz, C., ‘The theory of p-spaces with an application to convolution operators’, Trans. Amer Math. Soc. 154 (1971), 69–82.Google Scholar
[Jo1]Johnson, B. E., ‘Cohomology in Banach algebras’, Mem. Amer Math. Soc. 127 (Amer. Math. Soc., Providence, 1972).Google Scholar
[Jo2]Johnson, B. E., ‘Weak amenability of group algebras’, Bull. London Math. Soc. 23 (1991), 281–284.CrossRefGoogle Scholar
[La]Lau, A. T.-M., ‘The second conjugate algebra of the Fourier algebra of a locally compact group’, Trans. Amer Math. Soc. 267 (1981), 53–63.CrossRefGoogle Scholar
[LL]Lau, A. T.-M. and Loy, R. J., ‘Amenability of convolution algebras’, Math. Scand., to appear.Google Scholar
[LLW]Lau, A. T.-M., Loy, R. J. and Willis, G. A., ‘Amenability of Banach and C* algebras on locally compact groups’, Studia Math. 119 (1996), 161–178.Google Scholar
[Mc]McGehee, O. C., ‘Helson sets in T n’, in: Conference in Harmonic Analysis, College Park Maryland 1971, Lecture Notes in Math. 266 (Springer, Berlin, 1979) pp. 229–237.Google Scholar
[Ru1]Rudin, W., Fourier analysis on groups (Interscience, New York, 1960).Google Scholar
[Ru2]Rudin, W., Functional analysis (McGraw-Hill, New York, 1973).Google Scholar
[V]Varopoulos, N. Th., ‘Tensor algebras and harmonic analysis’, Acta Math. 119 (1967), 51–112.CrossRefGoogle Scholar