Multiplicatively and non-symmetric multiplicatively norm-preserving maps.

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let |.| and |.| denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. | Tf Tg| = | fg|, for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying | fg + α| = | Tf Tg + α|, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c( B) → c( A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c( B) such that for each f ∈ A, . In particular, if T satisfies the stronger condition R( fg + α) = R( Tf Tg + α), where R(.) denotes the peripheral range of algebra elements, then Tf( y) = T1( y) f( φ( y)), y ∈ c( B), for some homeomorphism φ: c( B) → c( A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying | Tf Tg| = | fg|, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations. [ABSTRACT FROM AUTHOR]