Linear Isometries between Real Banach Algebras of Continuous Complex-Valued Functions

Let and be compact Hausdorff spaces, and let and be topological involutions on and , respectively. In 1991, Kulkarni and Arundhathi characterized linear isometries from a real uniform function algebra on ( , ) onto a real uniform function algebra on ( , ) applying their Choquet boundaries and showed that these mappings are weighted composition operators. In this paper, we characterize all onto linear isometries and certain into linear isometries between and applying the extreme points in the unit balls of and . 1. Introduction and Preliminaries Let and denote the field of real and complex numbers, respectively. The symbol denotes a field that can be either or . The elements of are called scalars. We also denote by the set of all with . Let be a normed space over . We denote by and the dual space of and the closed unit ball of , respectively. For a subset of , let denote the set of all extreme points of . Kulkarni and Limaye showed [1, Theorem 2] that if is a nonzero linear subspace of and , then has an extension to some . We know that if and are normed spaces over and is a linear isometry from onto over , then is a bijection mapping between and . Let be a compact Hausdorff space. We denote by the unital commutative Banach algebra of all continuous functions from into , with the uniform norm , . We write instead as . For , we consider the linear functional on defined by ( ), which is called the evaluation functional on at . Clearly, for all . It is known [2, page 441] that Let be a real or complex linear subspace of . A nonempty subset of is called a boundary for (with respect to ), if for each the function assumes its maximum on at some . We denote by the intersection of all closed boundaries for . If is a boundary for , it is called the Shilov boundary for (with respect to ). Let be linear subspace of containing , the constant function with value on . A representing measure for is an -valued regular Borel measure on such that for all . Let . If , then and , the point mass measure on at , is a representing measure for . We denote by the set of all for which is the only representing measure for . If is a boundary for , it is called the Choquet boundary for (with respect to ). We know that Let be a real linear subspace of , and let be nonempty subset of . We say that is extremely regular at if for every open neighborhood of and for each there is a function with such that for all and for all . Let be a nonempty set. A self-map is called an involution on if for all . A subset of is called -invariant if . Clearly, if is -invariant, then . A -invariant