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arXiv · 2210.17131

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

David Cabezas·María Cueto-Avellaneda·Yuta Enami·Takeshi Miura·Antonio M. Peralta
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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 $|f(0)|+\|f'\|_\infty$.

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David Cabezas, María Cueto-Avellaneda, Yuta Enami, Takeshi Miura, Antonio M. Peralta. 2022-10-31. Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition. https://arxiv.org/abs/2210.17131

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