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Maximal algebras of continuous functions

1957; American Mathematical Society; Volume: 8; Issue: 1 Linguagem: Inglês

10.1090/s0002-9939-1957-0084732-8

1088-6826

Henry Helson, Frank Quigley,

Advanced Topology and Set Theory

in the topology of uniform convergence. In several recent papers John Wermer has considered some difficult special cases. The theorems presented here were suggested by certain of WVermer's results. Let S be the unit circle. Wermer has found a family of subalgebras of C(S) which are maximal among all closed subalgebras of C(S) [1; 2; 3]. These algebras can roughly be described in the following way: let a small copy of the circle be smoothly inscribed on a closed Riemann surface; consider the functions analytic on the part of the surface outside the circle and continuous on the circle as well. Functions of this type clearly form a closed subalgebra of C(S), which turns out to be maximal. This note is a beginning in the opposite direction. Given a compact space S and a maximal closed subalgebra { of C(S), we can prove that X shares a number of properties of algebras whose elements are analytic functions. Since we require no other hypotheses on S, this analysis can be carried only to a certain poinlt, and complete information for any particular space S will depend on subtler notions than we use. We assume, then, that C(S) = G is the algebra of continuous functions on the compact Hausdorff space S, and that 2f is a subalgebra of C, closed in the uniform topology and contained in no other proper closed subalgebra of d:. If it happens that W is an ideal in LY, it is clearly a maximal ideal and consists of all the continuous functions vanishing at a fixed point of S. (Conversely every maximal ideal is a maximal closed subalgebra.) We have in fact THEOREM 1. Either 2t is a maximal ideal in (S, or 2t contains the scalars.