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Maximality In Function Algebras

Published online by Cambridge University Press:  20 November 2018

Robert G. Blumenthal
Robert G. Blumenthal*
Affiliation:
University of Miami, Coral Gables, Florida
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In this paper we prove that the proper Dirichlet subalgebras of the disc algebra discovered by Browder and Wermer [1] are maximal subalgebras of the disc algebra (Theorem 2). We also give an extension to general function algebras of a theorem of Rudin [4] on the existence of maximal subalgebras of C(X). Theorem 1 implies that every function algebra defined on an uncountable metric space has a maximal subalgebra.

A function algebra A on X is a uniformly closed, point-separating subalgebra of C(X), containing the constants, where X is a compact Hausdorff space. If A and B are function algebras on X, A ⊂ B, A ≠ B, we say A is a maximal subalgebra of B if whenever C is a function algebra on X with A ⊂ C ⊂ B, either C = A or C = B.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 5 , 01 October 1970 , pp. 1002 - 1004
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Browder, A. and Wermer, J., A method for constructing Dirichlet algebras, Proc. Amer. Math. Soc. 15 (1964), 546–552.Google Scholar
2. Hoffman, K., Banach spaces of analytic functions (Prentice-Hall, Englewood Cliffs, N. J., 1962).Google Scholar
3. Pekzynski, A., Some linear topological properties of separable function algebras, Proc. Amer. Math. Soc. 18 (1967), 652–660.Google Scholar
4. Rudin, W., Subalgebras of spaces of continuous functions, Proc. Amer. Math. Soc. 7 (1956), 825–830.Google Scholar