BANACH FUNCTION ALGEBRAS AND CERTAIN POLYNOMIALLY NORM-PRESERVING MAPS

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively. Given a non-zero scalar alpha and s,t is an element of N we characterize the general form of suitable powers of surjective maps T, T' : A -> B satisfying parallel to(T f)(s)(T' g)(t) - alpha parallel to Y = parallel to f(s)g(t) - alpha parallel to X, for all f, g is an element of A, where parallel to.parallel to X and parallel to . parallel to y denote the supremum norms on X and Y, respectively. A similar result is given for the case where T = T' and T is defined between certain subsets of A and B. We also show that if T : A B is a surjective map, satisfying the stronger conditionR(pi) ((Tf)(s)(Tg)(t) - alpha) boolean AND R-pi(f(s) g(t) - alpha) not equal empty set for all f, g is an element of A, where R-pi(.) denotes the peripheral range of the algebra elements, then there exists a homeomorphism y from the Choquet boundary c(B) of B onto the Choquet boundary c(A) of A such that (T f)(d)(y) = (T1)(d)(y) (f o phi(y))(d) for all f E A and y E c(B),where d is the greatest common divisor of s and t.