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

Abstract

In 2022, Hatori [14] 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 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 ′ ∞.