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Weakly prime compact convex sets and uniform algebras

Published online by Cambridge University Press:  24 October 2008

A. J. Ellis
A. J. Ellis
Affiliation:
University College of Swansea
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Extract

1. Introduction. We introduce the notion of a weakly prime compact convex set, and we develop a reduction theory for spaces A(K). The notion is less restrictive in general than the prime compact convex sets of Chu, but gives a finer reduction than the Bishop and Silov decompositions forA(K) (12). The natural analogue for uniform algebras is related to the concept of weakly analytic sets due to Arenson, but unlike maximal weakly analytic sets the maximal weakly prime sets are always generalized peak sets; the uniform algebra can always be retrieved from the restrictions of the algebra to the maximal weakly prime sets.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 2 , March 1977 , pp. 225 - 232
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Alfsen, E. M.Compact convex sets and boundary integrals. Springer-Verlag, Berlin (1971).CrossRefGoogle Scholar
(2)Arenson, E. L.Certain properties of algebras of continuous functions. Soviet Math. Dokl. 7 (1966), 1522–1524.Google Scholar
(3)Arenson, E. L.Gleason parts and the Wermer-Hoffman theorem on characterizations of C(X). Siberian Math. J. 13 (1972), 831–838.CrossRefGoogle Scholar
(4)Arenson, E. L.An abstract Šilov-Bishop theorem. Optim. Vyp. 3 (20) (1971), 14–39. (M.R. 49. 3498.)Google Scholar
(5)Asimow, L.Directed Banach spaces of affine functions. Trans. Amer. Math. Soc. 143 (1969), 117–132.CrossRefGoogle Scholar
(6)Asimow, L.Decomposable compact convex sets and peak sets. for function spaces. Proc. Amer. Math. Soc. (1) 25 (1970), 75–79.Google Scholar
(7)Briem, E.Split faces associated with function algebras. J. London Math. Soc. (2) 9 (1975), 446–450.CrossRefGoogle Scholar
(8)Briem, E. Extreme orthogonal boundary measures for A(K) and decompositions for compact convex sets, Springer Lecture Notes in Mathematics 512 (1976), 8–16.CrossRefGoogle Scholar
(9)Chu, Cho-Ho. Anti-lattices and prime sets. Math. Scand. 31 (1972), 151–165.Google Scholar
(10)Ellis, A. J.On split faces and function algebras. Math. Ann. 195 (1972), 159–166.CrossRefGoogle Scholar
(11)Ellis, A. J.On facially continuous functions in function algebras. J. London Math. Soc. (2) 5 (1972), 561–564.CrossRefGoogle Scholar
(12)Ellis, A. J.Central decompositions and the essential set for the space A(K). Proc. London Math. Soc. (3) 26 (1973), 564–576.CrossRefGoogle Scholar
(13)Glicksbero, I.Measures orthogonal to algebras and sets of antisymmetry. Trans. Amer. Math. Soc. 105 (1962), 415–435.CrossRefGoogle Scholar
(14)Stout, E. L.The theory of uniform algebras. (Bogden and Quigley, Tarrytown on Hudson, New York 1971).Google Scholar