[Paper Review] Extending surjective isometries defined on the unit sphere of $\ell_\infty(\Gamma)$

Antonio M. Peralta|arXiv (Cornell University)|Sep 27, 2017

Advanced Banach Space Theory13 references4 citations

TL;DR

This paper proves that the Banach space ℓ∞(Γ), consisting of all complex-valued bounded functions on an infinite discrete set Γ, satisfies the Mazur-Ulam property: every surjective isometry between the unit spheres of ℓ∞(Γ) and an arbitrary complex Banach space X extends uniquely to a surjective real linear isometry from ℓ∞(Γ) onto X. The proof relies on geometric properties of partial isometries and a novel real-linear extension via a homogeneous product structure on ℓ∞(Γ).

ABSTRACT

Let $\Gamma$ be an infinite set equipped with the discrete topology. We prove that the space $\ell_{\infty}(\Gamma),$ of all complex-valued bounded functions on $\Gamma$, satisfies the Mazur-Ulam property, that is, every surjective isometry from the unit sphere of $\ell_{\infty}(\Gamma)$ onto the unit sphere of an arbitrary complex Banach space $X$ admits a unique extension to a surjective real linear isometry from $\ell_{\infty}(\Gamma)$ to $X$.

Motivation & Objective

• To establish that ℓ∞(Γ) satisfies the Mazur-Ulam property, extending a conjecture from prior work on c0(Γ).
• To resolve Tingley’s problem in the setting of complex Banach spaces by proving extension of isometries from the unit sphere of ℓ∞(Γ) to the whole space.
• To develop new geometric techniques based on partial isometries and M-orthogonality to overcome limitations of previous methods used for c0(Γ).
• To show that the homogeneous extension of a surjective isometry on S(ℓ∞(Γ)) is additive and real-linear, leading to a unique real linear extension.
• To confirm that ℓ∞(Γ) is the first known example of a complex Banach space satisfying the Mazur-Ulam property, beyond real sequence spaces.

Proposed method

• Introduce a real-linear product ⊙ on ℓ∞(Γ) such that λ∆(v) = ∆(λ ⊙v) for any non-zero partial isometry v and λ ∈𝕋 with |λ| = 1.
• Prove that a surjective isometry ∆ maps finite families of mutually orthogonal non-zero partial isometries in ℓ∞(Γ) to completely M-orthogonal sets in X.
• Use induction on the number of partial isometries to show that the homogeneous extension F satisfies F(∑ αj ⊙vj) = ∑ αj∆(vj) for complex coefficients αj.
• Leverage the norm density of algebraic elements (finite linear combinations of partial isometries) in ℓ∞(Γ) to extend additivity to all elements.
• Establish that the homogeneous extension F is Lipschitz and use approximation by finite combinations to prove F(a + b) = F(a) + F(b) for all a, b ∈ℓ∞(Γ).
• Conclude that F is a real linear isometry extending ∆, and uniqueness follows from the isometry property on the unit sphere.

Experimental results

Research questions

• RQ1 Does the space ℓ∞(Γ) satisfy the Mazur-Ulam property, i.e., can every surjective isometry between its unit sphere and another complex Banach space’s unit sphere be uniquely extended to a real linear isometry?
• RQ2 Can the geometric structure of partial isometries in ℓ∞(Γ) be used to characterize the behavior of isometries on the unit sphere?
• RQ3 Is it possible to define a real-linear product ⊙ on ℓ∞(Γ) such that the isometry ∆ respects scalar multiplication by unimodular complex numbers via this product?
• RQ4 How does the complete M-orthogonality of images of orthogonal partial isometries in X relate to the additivity of the homogeneous extension?
• RQ5 Can the additivity of the homogeneous extension be established via approximation by finite sums of partial isometries and Lipschitz continuity?

Key findings

• The space ℓ∞(Γ) satisfies the Mazur-Ulam property: every surjective isometry from S(ℓ∞(Γ)) onto S(X) for any complex Banach space X extends uniquely to a surjective real linear isometry from ℓ∞(Γ) onto X.
• The homogeneous extension F of a surjective isometry ∆: S(ℓ∞(Γ)) → S(X) is additive on ℓ∞(Γ), i.e., F(a + b) = F(a) + F(b) for all a, b ∈ℓ∞(Γ).
• For any finite family of mutually orthogonal non-zero partial isometries {v₁, ..., vₖ} in ℓ∞(Γ), their images under ∆ form a completely M-orthogonal set in X.
• The extension F satisfies F(∑ αj ⊙vj) = ∑ αj∆(vj) for complex coefficients αj, where ⊙ is a real-linear product on ℓ∞(Γ) compatible with unimodular scalars.
• The real-linear structure of the extension is established via approximation by algebraic elements (finite sums of partial isometries), which are norm-dense in ℓ∞(Γ).
• The extension is unique and isometric, and the proof relies on novel geometric arguments involving partial isometries and M-orthogonality, distinct from earlier methods used for c0(Γ).