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

2022/10/31 by Cabezas, David, Cueto-Avellaneda, María, Enami, Yuta +2
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2210.17131

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 C¹([0,1]) of all continuously differentiable complex-valued functions on [0,1] equipped with the norm |f(0)|+‖f'‖_∞.

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