The Mazur-Ulam property for a Banach space which satisfies a separation condition

Abstract

We study C-rich spaces, lush spaces, and C-extremely regular spaces concerning with the Mazur-Ulam property. We show that a uniform algebra and 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 complex-valued continuous functions on a locally compact Hausdorff space which vanish at infinity is C-extremely regular provided that it separates the points of the underlying space and has no common zeros. In section 3 we exhibit descriptions on the Choquet bounday, the Šilov bounday, the strong boundary points. We also recall the definition that a function space strongly separates the points in the underlying space. We need to avoid the confusion which appears because of the variety of names of these concepts; they sometimes differs from authors to authors. After some preparation, we study the Mazur-Ulam property in sections 4 through 6. 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 section 5 we consider a Banach space with a separation condition ( * ) (Definition 5.1). We prove that a real Banach space satisfying ( * ) has the Mazur-Ulam propety (Theorem 6.1), and a complex Banach space satisfying ( * ) has the complex Mazur-Ulam property (Theorem 6.3). Applying the results in the previous sections we prove that an extremely Cregular complex linear subspace has the complex Mazur-Ulam property (Corollary 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).

Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

“…Many mathematicians have been involved in the study of the Mazur-Ulam property for several Banach spaces (cf. [3,4,6,14,16,26,27]). A refinement of the Mazur-Ulam property was introduced by Hatori in [14].…”

“…for all λ ∈ T and η ∈ Ch(A) × T. Now we introduce the Hausdorff distance of maximal convex subsets in S B , which is used in [14], [16] and [20] to solve Tingley's problem.…”

“…Since A is extremely C-regular, there exists u 1 ∈ S A such that u 1 (x 1 ) = 1 and u 1 (x 2 ) = −1 (see [16,Lemma 5.4]). Let us define…”