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Given an infinite set Γ, we prove that the space of complex null sequences c 0 (Γ) satisfies the Mazur-Ulam property, that is, for each Banach space X, every surjective isometry from the unit sphere of c 0 (Γ) onto the unit sphere of X admits a (unique) extension to a surjective real linear isometry from c 0 (Γ) to X. We also prove that the same conclusion holds for the finite dimensional space ℓ m ∞ .2010 Mathematics Subject Classification. Primary 46B20, 46A22, 46B04, 46B25.

We show that the isometry groups of Lip( X, d) and lip( X, d α ) with α ∈ (0, 1), for a compact metric space (X, d), are algebraically reflexive. We also prove that the sets of isometric reflections and generalized bi-circular projections on such spaces are algebraically reflexive. In order to achieve this, we characterize generalized bi-circular projections on these spaces.

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