Spectral Conditions for Composition Operators on Algebras of Functions

We establish general sufficient conditions for maps between function algebras to be composition or weighted composition operators, which extend previous results in [2, 4, 6, 7]. Let X be a locally compact Hausdorff space and A subset of C(X) a dense subalgebra of a function algebra, not necessarily with unit, such that X = DA and p(A) = delta A, where partial derivative A is the Shilov boundary, partial derivative A the Choquet boundary, and p(A) the set of p -points of A. If T: A -> B is a surjective map onto a function algebra B subset of C(Y) such that either sigma(pi),(T f . Tg) subset of sigma(pi),(f g) for all f, g is an element of A, or, alternatively, sigma(pi),(f g) subset of sigma(pi),(T f . Tg) for all f, g is an element of A, then there is a homeomorphism psi: delta B -> delta A and a function a on delta B so that (T f)(y) = a(y) f (psi(y)) for all f is an element of A and y is an element of delta B. If, instead, sigma(pi),(T f . Tg) boolean AND sigma(pi),(f g) not equal phi for all f, g is an element of A, and either sigma(pi),(f) subset of sigma(pi),(T f) for all f is an element of A, or, alternatively, sigma(pi),(T f) subset of sigma(pi),(f) for all f is an element of A, then (T f)(y) = f (psi(y)) for all f is an element of A and y is an element of delta B. In particular, if A and B are uniform algebras and T : A -> B is a surjective map with sigma(pi), (T f . T g) boolean AND sigma(pi), (f g) not equal phi for all f, g is an element of A, that has a limit, say b, at some a e A with a(2) = 1, then (T f)(y) = b(y)alpha(psi (y)) f (is an element of(y)) for every f is an element of A and y is an element of delta B.