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The Maximal Ideal Space of Subalgebras of the Disk Algebra

Published online by Cambridge University Press:  20 November 2018

Bruce Lund
Bruce Lund*
Affiliation:
Mathematics Department, University of New Brunswick Fredericton, N.B., Canada
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Let X be a compact Hausdorff space and C(X) the complexvalued continuous functions on X. We say A is a function algebra on X if A is a point separating, uniformly closed subalgebra of C(X) containing the constant functions. Equipped with the sup-norm ‖f‖ = sup{|f(x)|: x ∊ X} for f ∊ A, A is a Banach algebra. Let MA denote the maximal ideal space.

Let D be the closed unit disk in C and let U be the open unit disk. We call A(D)={f ∊ C(D):f is analytic on U} the disk algebra. Let T be the unit circle and set C1(T) = {f ∊ C(T): f'(t) ∊ C(T)}.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 1 , April 1975 , pp. 61 - 65
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Bjork, J.-E., Analytic structures in the maximal ideal space of a uniform algebra, Ark. Mat. 8 (1970), 239-244.Google Scholar
2. Bjork, J.-E., Holomorphic convexity and analytic structures in Banach algebras, Ark. Mat. 9 (1971), 39-54.Google Scholar
3. Gamelin, T. W., Polynomial approximation on thin sets, Symposium on Several Complex Variables, Park City, Utah, 1970. Springer-Verlag, Heidelberg (1971), 50-78.Google Scholar
4. Glicksberg, I., A remark on analyticity of function algebras, Pacific J. Math. 13 (1963), 1181-1185.Google Scholar
5. Goluzin, G. M., Geometric Theory of Functions of a Complex Variable, 26, Translations of Mathematical Monographs. American Math Society, Providence, R.I., 1969.Google Scholar
6. Lund, B., Ideals and subalgebras of a function algebra, Canad. J. Math., 26, (1974), 405-411.Google Scholar
7. Newman, M. H. A., Topology of Plane Sets. Cambridge University Press, Cambridge, U.K., 1954.Google Scholar
8. Novinger, W. P., Holomorphic functions with infinitely dijferentiable boundary values, Ill. J. Math. 15 (1971), 80-90.Google Scholar
9. Stout, E., The Theory of Uniform Algebras. Bogden and Quigley, Tarry town-on-Hudson, 1971.Google Scholar