Orthogonally additive polynomials on Banach function algebras

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(fd) (f∈A) for some continuous linear map T:A→X for a variety of Banach function algebras such as the algebra of continuous functions C0(Ω) for any locally compact Hausdorff space Ω, the algebra of Lipschitz functions lipα(K) for any compact metric space K and α∈]0,1[, the Figà-Talamanca-Herz algebra Ap(G) for some locally compact groups G and p∈]1,+∞[, the algebras AC([a,b]) and BVC([a,b]) of absolutely continuous functions and of continuous functions of bounded variation on the interval [a,b]. In the case where A=Cn([a,b]), P can be represented as P(f)=∑T(n1,…,nd)(f(n1)⋯f(nd)), where the sum is taken over (n1,…,nd)∈Zd with 0≤n1≤…≤nd≤n, for appropriate continuous linear maps T(n1,…,nd):Cn−nd([a,b])→X.