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

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 \vSilov 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. 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 $C$-regular 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).