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We construct certain Banach algebras of infinitely differentiable functions on compact plane sets such that the algebras are quasianalytic, and we use these algebras to construct examples of Banach algebras defined on their maximal ideal spaces which, first, have only countably many peak points and, second, have the property that a discontinuous function operates on the algebra. We show that any function defined on an open subset of the plane which operates on a Banach function algebra is necessarily continuous on a dense open subset of its domain.

In this paper we study the functions which operate on Re A (the space of real parts of elements of a Banach function algebra A). We prove as our main result that if A is uniform or if A is ultraseparating, then only linear functions operate boundedly. We finally obtain a dichotomy for symbolic calculus on C̃(K), where K is a compact subset of the unit circle T andC̃(K) denotes the space of those continuous functions ƒ on K which admit a continuous extension to T whose Fourier conjugate is continuous.