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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 senseWe 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.

Abstract. In this paper we describe into real-linear isometries defined between (not necessarily unital) function algebras and show, based on an example, that this type of isometries behave differently from surjective real-linear isometries and classical linear isometries. Next we introduce jointly norm-additive mappings and apply our results on real-linear isometries to provide a complete description of these mappings when defined between function algebras which are not necessarily unital or uniformly closed.

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖X and ‖.‖Y 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‖Y = ‖fg‖X, 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‖X = ‖Tf Tg + α‖Y, 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, $$ Tf\left( y \right) = \left\{ \begin{gathered} \eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\ - \frac{\alpha } {{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\ \end{gathered} \right. $$. In particular, if T satisfies the stronger condition R π(fg + α) = R π(Tf Tg + α), where R π(.) 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‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations.

Abstract. Let A 1 , ..., A k be function algebras (or more generally, dense subspaces of uniformly closed function algebras) on locally compact Hausdorff spaces X 1 , ..., X k , respectively, and let Z be a locally compact Hausdorff space. A k-linear map T :Based on a new version of the additive Bishop's Lemma, we provide a weighted composition characterization of such maps. These results generalize the well-known Holsztyński's theorem ([9]) and the bilinear version of this theorem provided in [10] by a different approach.