The Mazur-Ulam property for a Banach space which satisfies a separation condition (Research on preserver problems on Banach algebras and related topics)

Abstract

After some preparations in section 1, we recall the three well known concepts: the Choquet boundary, the Šilov boundary, and the strong boundary points in section 2. We need to define them by avoiding the confusion which appears because of the variety of names of these concepts; they sometimes differs from authors to authors. We describe the relationship between the three concepts emphasizing the case where a function space strongly separates the points in the underlying space. We study C-rich spaces, lush spaces, and extremely C-regular spaces concerning with the Mazur-Ulam property in section 3. We show that a uniform algebra and the uniform closure of the real part of a uniform algebra with the supremum norm are C-rich spaces, hence lush spaces. We prove that a uniformly closed subalgebra of the algebra of all complex-valued continuous functions on a locally compact Hausdorff space which vanish at infinity is extremely C-regular provided that it separates the points of the underlying space and has no common zeros. We exhibit a space of harmonic functions which has the Mazur- Ulam property (Corollary 3.8). The main concern in sections 4 through 6 is the Mazur-Ulam property. We exhibit a sufficient condition on a Banach space which has the Mazur-Ulam property and the complex Mazur-Ulam property (Propositions 4.11 and 4.12). In sections 5 and 6 we consider a Banach space with a separation condition (∗) (Definition 5.1). We prove that a real Banach space satisfying (∗) has the Mazur-Ulam property (Theorem 6.1), and a complex Banach space satisfying (∗) has the complex Mazur-Ulam property (Theorem 6.3). Applying these theorems and the results in the previous sections we prove that an extremely C-regular real (resp. complex) linear subspace has the (resp. complex) Mazur-Ulam property (Corollary 6.2 (resp. 6.4)) in section 6. As a consequence we prove that any closed subalgebra of the algebra of all complex-valued continuous functions defined on a locally compact Hausdorff space has the complex Mazur-Ulam property (Corollary 6.5).