Multiplicatively range-preserving maps between Banach function algebras

Let A and B be two Banach function algebras on locally compact Hausdorff spaces X and Y, respectively. Let T be a multiplicatively range-preserving map from A onto B in the sense that ( T f T g ) ( Y ) = ( f g ) ( X ) for all f , g ∈ A . We define equivalence relations on appropriate subsets X ˜ and Y ˜ of X and Y, respectively, and show that T induces a homeomorphism between the quotient spaces of X ˜ and Y ˜ by these equivalence relations. In particular, if all points in the Choquet boundaries of A and B are strong boundary points, then X ˜ and Y ˜ are equal to the Choquet boundaries of A and B, respectively, and moreover, there exist a continuous function h on the Choquet boundary of B taking its values in { − 1 , 1 } and a homeomorphism φ from the Choquet boundary of B onto the Choquet boundary of A such that T f ( y ) = h ( y ) f ( φ ( y ) ) for all f ∈ A and y in the Choquet boundary of B. For certain Banach function algebras A and B on compact Hausdorff spaces X and Y, respectively, we can weaken the surjectivity assumption and give a representation for maps belonging 2-locally to the family of all multiplicatively range-preserving maps from A onto B.