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

We aim to generalize Figiel’s theorem to the local unit spheres case. Let T : S E ⟶ 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 ( ∗ ) cov { - T ( C ) ∪ T ( - C ) } ⊂ S F for every maximal convex subset C of S E 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.