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Amenability of Banach algebras

Published online by Cambridge University Press:  24 October 2008

Frédéric Gourdeau
Frédéric Gourdeau
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB
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Extract

We consider the problem of amenability for a commutative Banach algebra. The question of amenability for a Banach algebra was first studied by B. E. Johnson in 1972, in [5]. The most recent contributions, to our knowledge, are papers by Bade, Curtis and Dales [1], and by Curtis and Loy [3]. In the first, amenability for Lipschitz algebras on a compact metric space K is studied. Using the fact, which they prove, that LipαK is isometrically isomorphic to the second dual of lipαK, for 0 < α < 1, they show that lipαK is not amenable when K is infinite and 0 < α < 1. In the second paper, the authors prove, without using any serious cohomology theory, some results proved earlier by Khelemskii and Scheinberg [8] using cohomology. They also discuss the amenability of Lipschitz algebras, using the result that a weakly complemented closed two-sided ideal in an amenable Banach algebra has a bounded approximate identity. Their result is stronger than that of [1].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 2 , March 1989 , pp. 351 - 355
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Bade, W. G., Curtis, P. C. Jr. and Dales, H. G.. Amenability and weak-amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc. (3) 55 (1987), 359–377.CrossRefGoogle Scholar
[2]Bonsall, F. F. and Duncan, J.. Complete normed algebras (Springer-Verlag, 1973).CrossRefGoogle Scholar
[3]Curtis, P. C. Jr. and Loy, R. J.. The structure of amenable Banach algebras. (To appear.)Google Scholar
[4]Grønbæk, N.. Amenability of Weighted Discrete Convolution Algebras on Cancellative Semigroups. (To appear.)Google Scholar
[5]Johnson, B. E.. Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 (1972).Google Scholar
[6]Johnson, B. E.. Perturbations of Banach algebras. Proc. London Math. Soc. (3) 34 (1977), 439–458.CrossRefGoogle Scholar
[7]Johnson, J. A.. Banach spaces of Lipschitz functions and vector-valued Lipschitz functions. Trans. Amer. Math. Soc. 148 (1970), 147–169.CrossRefGoogle Scholar
[8]Khelemskii, A. Ya.. Flat Banach modules and amenable algebras. Trans. Moscow Math. Soc. 47 (1984), 199–224.Google Scholar
[9]Sherbert, D.. Banach algebras of Lipschitz functions. Pacific J. Math. 13 (1963), 1387–1399.CrossRefGoogle Scholar