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The Slice Approximating Property and Figiel-Type Problem on Unit Spheres

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Abstract

We aim to generalize Figiel’s theorem to the local unit spheres case. Let \(T:S_E\longrightarrow S_F\) be an isometric embedding (not necessarily surjective) between unit spheres of Banach spaces. Does T admit the Figiel operator? Recently, some attempts were made by Liu and Yin (Acta Math Sci 43B(4):1503–1517, 2023), and a counterexample was given to this question, which shows that Figiel’s theorem cannot be routinely generalized to the unit spheres case. After that, a necessary condition

$$\begin{aligned} \begin{aligned}{}&{} (*)\; \text {cov} \{-T(C)\cup T(-C)\}\subset S_F \text{ for } \text{ every } \text{ maximal } \text{ convex } \text{ subset } C \text { of } S_E \end{aligned} \end{aligned}$$

was obtained for the existence of the Figiel operator. Naturally, Liu and Yin proposed the reformulated Figiel-type problem: is the existence of the Figiel operator for an isometric embedding T equivalent to the condition \((*)\)? They answered this problem affirmatively in the case that E is a space with the Tingley property (T-property for short). To attack this problem, we introduce the concept of the slice approximating property (SAP for short) and show that all uniformly convex spaces, almost CL-spaces, and spaces with the T-property admit the SAP. Furthermore, we give an affirmative answer to the reformulated Figiel-type problem in the case that E is a local GL space with the SAP. This generalizes one of the main results in Liu and Yin (Acta Math Sci 43B(4):1503–1517, 2023) to a much more extensive case. At the end of this paper, an interesting result about isometric embedding and dimensions is proved.

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Funding

R. Liu was partially supported by the National Natural Science Foundation of China (No.11671214, 11971348, and 12071230). L. Li was partially supported by the National Natural Science Foundation of China (No.12171251).

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Lei Li & Rui Liu

  2. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453000, China

    Jifu Yin

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  1. Lei Li
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  2. Rui Liu
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  3. Jifu Yin
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Correspondence to Jifu Yin.

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Li, L., Liu, R. & Yin, J. The Slice Approximating Property and Figiel-Type Problem on Unit Spheres. Results Math 79, 40 (2024). https://doi.org/10.1007/s00025-023-02066-3

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