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.