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Translation invariance, compact semigroups and injective coalgebras

Published online by Cambridge University Press:  24 October 2008

Andrew M. Tonge
Andrew M. Tonge
Affiliation:
Université de Paris-Sud, Orsay, France; St John's College, Cambridge
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Extract

The lack of a satisfactory notion of the dual of a compact semigroup S precludes a simple concrete characterization of the closed translation invariant subspaces of C(S). In this article we develop an abstract theory of such spaces (satisfying a weak version of the Banach approximation property) in terms of ‘injective coalgebras’. We characterize their dual spaces as those Banach algebras whose closed unit balls are compact semigroups under the weak star topology. Injective Hopf algebras are shown to be the spaces C(G), G a compact group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 2 , March 1978 , pp. 225 - 236
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Davie, A. M.The approximation property for Banach spaces. Bull. London Math. Soc. 5 (1973), 261–266.CrossRefGoogle Scholar
(2)Grothendieck, A.Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16 (1955).Google Scholar
(3)Grothendieck, A.Résumé de la théorie inétrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8 (1956), 1–79.Google Scholar
(4)Hewitt, E. and Ross, K. A.Abstract harmonic analysis, vol. II (Springer, 1970).Google Scholar
(5)Kaijser, S.Some remarks on injective Banach algebras. Springer Lecture Notes in Mathe-matics, 512 (1976), 84–95.Google Scholar
(6)Sweedler, M. E.Hopf Algebras. Benjamin (1969).Google Scholar
(7)Tonge, A. M. La presque périodicité et les coalgebres injectives. Studia Math. (in the press).Google Scholar
(8)Varopoulos, N.Some remarks on Q-algebras. Ann. Inst. Fourier Grenoble 22 (1972), 1–11.CrossRefGoogle Scholar
(9)Varopoulos, N.Sur les quotients d' algèbres uniformes. C. R. Acad. Sci., Paris 274 (1972), 1344–1346.Google Scholar