  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).
