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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.

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We study holomorphic maps between C * -algebras A and B. When f : BA(0, ̺) −→ B is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball U = BA(0, δ) and we assume that f is orthogonality preserving on Asa ∩ U , orthogonally additive on U and f (U ) contains an invertible element in B, then there exist a sequence (hn) in B * * and Jordan * -homomorphisms Θ, Θ :Θ(a n )hn, uniformly in a ∈ U . When B is abelian the hypothesis of B being unital and f (U ) ∩ inv(B) = ∅ can be relaxed to get the same statement.2010 MSC: Primary 46G20, 46L05; Secondary 46L51, 46E15, 46E50.

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