The Mazur-Ulam property for the space of complex null sequences

Extending surjective isometries defined on the unit sphere of $\ell_\infty(Γ)$

“…The main result in [12] provides what is probably the first example of a complex Banach space satisfying the Mazur-Ulam property. Prior contributions showed that the spaces c 0 (Γ, R), of real null sequences, and ℓ ∞ (Γ, R), of all bounded real-valued functions on a discrete set Γ, satisfy the Mazur-Ulam property (see [6,Corollary 2], [14,Main Theorem]).…”

“…We consider c 0 (Γ) as a Banach space equipped with the supremum norm. In a recent contribution, we prove that the space c 0 (Γ) satisfies the Mazur-Ulam property, that is, every surjective isometry from the unit sphere of c 0 (Γ) to the unit sphere of an arbitrary complex Banach space X admits a unique extension to a surjective real linear isometry from c 0 (Γ) onto X (see [12]).…”

“…Another examples of Banach spaces satisfying the Mazur-Ulam property are the space C(K, R), of all real-valued continuous functions on a compact Hausdorff space K [14, Corollary 6], and the spaces L p ((Ω, Σ, µ), R) of real-valued measurable functions on an arbitrary σ-finite measure space (Ω, Σ, µ) for all 1 ≤ p ≤ ∞ [17,16,18]. Although, a surjective linear isometry between the unit spheres of two complex Banach spaces need not admit an extension to a complex linear surjective isometry between the spaces (consider, for example, the conjugation on S(C)), the recent contributions on Tingley's problem for (complex) sequence spaces and operator algebras (compare [19,20,21,7,9,11] and the recent reference [12]) show the interest and attractiveness of the study of the Mazur-Ulam property in the setting of complex Banach spaces.…”

“…It is conjectured in [12] that the space ℓ ∞ (Γ) also satisfies the Mazur-Ulam property. However, the techniques in the just quoted paper are not enough to prove this affirmation.…”

“…The strategy in this note is to improve the geometric study developed in [12] to establish that c 0 (Γ) satisfies the Mazur-Ulam property. Here, instead of considering minimal projections in c 0 (Γ), we consider general non-zero partial isometries in ℓ ∞ (Γ).…”