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Continuation of analytic structure in the maximal ideal space of a uniform algebra

Published online by Cambridge University Press:  24 October 2008

John R. Shackell
John R. Shackell
Affiliation:
University of Kent
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Let A be a uniform algebra with maximal ideal space M and Shilov boundary Σ; see (8), (18) or (21) for the basic definitions. If Σ is different from M, there is often analytic structure in M/Σ. However this is not always the case, as is shown by the classical example of Stolzenberg in (16). Hence much of the considerable amount of research on this topic has been devoted to finding conditions which ensure the presence of analytic structure in M/Σ One particularly fruitful line of development has been concerned with one-dimensional analytic structure; in particular we have in mind the classical theorem of Bishop (see (2), chapter 11) and the more recent result of Aupetit and Wermer(2).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 3 , November 1982 , pp. 437 - 449
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Allan, G. R.A note on the holomorphic functional calculus in a Banach algebra. Proc. Amer. Math. Soc. 22 (1969), 77–81.CrossRefGoogle Scholar
(2)Aupetit, B. and Wermer, J.Capacity and uniform algebras. J. Funct. Anal. 28 (1978), 386–400.CrossRefGoogle Scholar
(3)Basener, R. F.A generalized Shilov boundary and analytic structure. Proc. Amer. Math. Soc. 47 (1975), 98–104.CrossRefGoogle Scholar
(4)Bear, H. S. and Hile, G. N.Analytic structure in function algebras. Houston J. Math. 5 (1979), 21–28.Google Scholar
(5)Bredon, G. E.Sheaf theory (McGraw-Hill, 1967).Google Scholar
(6)Clifford, A. H. and Preston, G. B.The algebraic theory of semigroups, vol. 1 (American Mathematical Society, Providence, R.I., 1961).Google Scholar
(7)Craw, I. G.A condition equivalent to the continuity of characters on a Fréchet algebra. Proc. London Math. Soc. (3) 22 (1971), 452–464.Google Scholar
(8)Gamelin, T. W.Uniform algebras (Prentice Hall, Englewood Cliffs, N.J., 1969).Google Scholar
(9)Gunning, R. C. and Rossi, H.Analytic functions of several complex variables (Prentice Hall, Englewood Cliffs, N.J., 1965).Google Scholar
(10)Rickart, C. E.Analytic phenomena in general function algebras. Pacific J. Math. 18 (1960), 361–377.Google Scholar
(11)Rickart, C. E.Analytic functions of an infinite number of complex variables. Duke Math. J. 36 (1969), 581–598.Google Scholar
(12)Rickart, O. E.Natural function algebras (Springer-Verlag, N.Y., 1979).CrossRefGoogle Scholar
(13)Shackell, J. R.Semigroups associated with polynomial hull structure. Proc. London Math. Soc. (3) 39 (1979), 30–50.CrossRefGoogle Scholar
(14)Sibony, N. Multidimensional analytic structure in the spectrum of a uniform algebra. Spaces of analytic functions, Kristiansand, Norway, Springer Lecture Notes, no. 512 (1975), 139–165.Google Scholar
(15)Stolzenberg, G.An example concerning rational convexity. Math. Ann. 147 (1962), 275–276.Google Scholar
(16)Stolzenberg, G.A hull with no analytic structure. J. Math. Mech. 12 (1963), 103–111.Google Scholar
(17)Stolzenberg, G.Polynomially and rationally convex sets. Acta Math. 109 (1963), 259–289.Google Scholar
(18)Stout, E. L.The theory of uniform algebras (Bodgen and Quigley, N.Y., 1971).Google Scholar
(19)Taylor, J. L.Topological invariants of the maximal ideal space of a Banach algebra. Advances in Math. 19 (1976), 149–206.CrossRefGoogle Scholar
(20)Wermer, J.An example concerning polynomial convexity. Math. Ann. 139 (1959), 147–150.Google Scholar
(21)Wermer, J.Banach algebras and several complex variables, 2nd ed. (Springer-Verlag, N.Y., 1976).Google Scholar