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Localization for uniform algebras generated by smooth functions on two‐manifolds

Conditions are given under which a uniform algebra on a two‐manifold must contain all continuous functions. In particular, it is shown that, if A is a uniform algebra generated by smooth functions on a compact smooth two‐manifold M such that the maximal ideal space of A is M, and every continuous function on M is locally a uniform limit of functions in A, then A = C(M). This gives an affirmative answer to a special case of a question from the Proceedings of the Symposium on Function Algebras held at Tulane University in 1965. It is also shown that, on every smooth manifold of dimension at least four, there exists a uniform algebra that is nonlocal. Additional related results are also established.