Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

TL;DR

This paper identifies a class of complex Banach spaces, including Lipschitz and differentiable function spaces, where surjective isometries on the unit sphere extend to the entire space, despite not satisfying the Hausdorff distance condition.

Contribution

It introduces a new class of complex Banach spaces that satisfy the Mazur--Ulam property without meeting the Hausdorff distance condition, expanding understanding of isometry extensions.

Findings

Surjective isometries on the unit sphere extend to the whole space for the new class.

Includes spaces like Lip([0,1]) and C^1([0,1]) with specific norms.

Provides examples where the Hausdorff distance condition is not satisfied but the property still holds.

Abstract

In 2022, Hatori gave a sufficient condition for complex Banach spaces to have the complex Mazur--Ulam property. In this paper, we introduce a class of complex Banach spaces $B$ that do not satisfy the condition but enjoy the property that every surjective isometry on the unit sphere of such $B$ admits an extension to a surjective real linear isometry on the whole space $B$. Typical examples of Banach spaces studied in this note are the spaces ${\rm Lip}([0,1])$ of all Lipschitz complex-valued functions on $[0,1]$ and $C^1([0,1])$ of all continuously differentiable complex-valued functions on $[0,1]$ equipped with the norm $|f(0)|+\|f'\|_\infty$.