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Character amenability of Banach algebras

Published online by Cambridge University Press:  01 May 2008

MEHDI SANGANI MONFARED
MEHDI SANGANI MONFARED*
Affiliation:
Department of Mathematics and Statistics, University of Windsor, Windsor, ON, N9B 3P4, Canada. e-mail: monfared@uwindsor.ca
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Abstract

We introduce the notion of character amenable Banach algebras. We prove that character amenability for either of the group algebra L1(G) or the Fourier algebra A(G) is equivalent to the amenability of the underlying group G. Character amenability of the measure algebra M(G) is shown to be equivalent to G being a discrete amenable group. We also study functorial properties of character amenability. For a commutative character amenable Banach algebra A, we prove all cohomological groups with coefficients in finite-dimensional Banach A-bimodules, vanish. As a corollary we conclude that all finite-dimensional extensions of commutative character amenable Banach algebras split strongly.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 3 , May 2008 , pp. 697 - 706
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Bade, W. G., Dales, H. G. and Lykova, Z. A.. Algebraic and strong splittings of extensions of Banach algebras. Mem. Amer. Math. Soc. Vol. 656 (Amer. Math. Soc., 1999).Google Scholar
[2]Curtis, P. C. and Loy, R. J.. The structure of amenable Banach algebras. J. London Math. Soc. 40, (1989), 89–104.CrossRefGoogle Scholar
[3]Dales, H. G.. Banach Algebras and Automatic Continuity. (Oxford Uinversity Press, 2000).Google Scholar
[4]Dales, H. G., Ghahramani, F. and Helemskii, A. Ya.. The amenability of measure algebras. J. London Math. Soc. 66 (2002), 213–226.CrossRefGoogle Scholar
[5]Eymard, P.. L'algèbre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92 (1964), 181–236.CrossRefGoogle Scholar
[6]Eymard, P.. Algèbres Ap et convoluteurs de Lp. Lecture Notes in Math. Vol. 180, pp. 364–381 (Springer-Verlag, 1971).Google Scholar
[7]Figà–Talamanca, A.. Translation invariant operators in Lp. Duke Math. J. 32 (1965), 459–501.CrossRefGoogle Scholar
[8]Forrest, B., Kaniuth, E., Lau, A. and Spronk, N.. Ideals with bounded approximate identities in Fourier algebras. J. Funct. Anal. 203 (2003), 268–304.CrossRefGoogle Scholar
[9]Granirer, E. E.. On some spaces of linear functionals on the algebra Ap(G) for locally compact groups. Colloq. Math. LII (1987), 119–132.CrossRefGoogle Scholar
[10]Helemskii, A. Ya.. A certain class of flat Banach modules and its applications. Vest. Mosk. Univ. Ser. Mat. Mekh. 27 (1972), 29–36.Google Scholar
[11]Helemskii, A. Ya. Flat Banach modules and amenable Banach algebras. Trans. Moscow Math. Soc. 47 (1985), 229–250.Google Scholar
[12]Helemskii, A. Ya.. The Homology of Banach and Topological Algebras (Kluwer Academic Publishers, 1989).CrossRefGoogle Scholar
[13]Herz, C.. Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23 (1973), 91–123.CrossRefGoogle Scholar
[14]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis, Vol. 2 (Springer-Verlag, 1970).Google Scholar
[15]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis, Vol. 1, Second edition (Springer-Verlag, 1979).CrossRefGoogle Scholar
[16]Johnson, B. E.. Cohomology in Banach Algebras. Mem. Amer. Math. Soc. Vol. 127 (Amer. Math. Soc., 1972).Google Scholar
[17]Kaniuth, E., Lau, A. T. and Pym, J.. On φ-Amenability of Banach algebras. Math. Proc. Camb. Phil. Soc. (to appear).Google Scholar
[18]Lau, A. T.-M.. Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups. Fund. Math. 118 (1983), 161–175.Google Scholar
[19]Runde, V.. Lectures on Amenability (Springer, 2002).CrossRefGoogle Scholar
[20]Sangani–Monfared, M.. Extensions and isomorphims for the generalized Fourier algebras of a locally compact group. J. Funct. Anal. 198 (2003), 413–444.CrossRefGoogle Scholar
[21]Sangani–Monfared, M.. On certain products of Banach algebras with applications to harmonic analysis. Studia Math. 178 (2007), 277–294.CrossRefGoogle Scholar
[22]Steiniger, H.. Finite-dimensional extensions of the Fourier algebras, (preprint).Google Scholar