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For a Banach function algebra A, we consider the problem of representing a continuous d-homogeneous polynomial P : A → X , where X is an arbitrary Banach space, that satisfies the property P ( f + g ) = P ( f ) + P ( g ) whenever f , g ∈ A are such that supp ( f ) ∩ supp ( g ) = ∅ . We show that such a polynomial can be represented as P ( f ) = T ( f d ) ( f ∈ A ) for some continuous linear map T : A → X for a variety of Banach function algebras such as the algebra of continuous functions C 0 ( Ω ) for any locally compact Hausdorff space Ω, the algebra of Lipschitz functions lip α ( K ) for any compact metric space K and α ∈ ] 0 , 1 [ , the Figa–Talamanca–Herz algebra A p ( G ) for some locally compact groups G and p ∈ ] 1 , + ∞ [ , the algebras A C ( [ a , b ] ) and B V C ( [ a , b ] ) of absolutely continuous functions and of continuous functions of bounded variation on the interval [ a , b ] . In the case where A = C n ( [ a , b ] ) , P can be represented as P ( f ) = ∑ T ( n 1 , … , n d ) ( f ( n 1 ) ⋯ f ( n d ) ) , where the sum is taken over ( n 1 , … , n d ) ∈ Z d with 0 ≤ n 1 ≤ … ≤ n d ≤ n , for appropriate continuous linear maps T ( n 1 , … , n d ) : C n − n d ( [ a , b ] ) → X .

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.