The Figiel type 问题 on unit spheres

Abstract

In this paper, we study the Figiel type 问题 of isometries between the unit spheres of Banach spaces. First, by a counter例, we show that the classical Figiel's 定理 cannot be trivially generalized to the case of isometric embeddings on unit spheres. Then we find a natural necessary condition to ensure the corresponding Figiel type 定理. Under this natural condition, we prove the Figiel type theorems of isometries from the unit sphere of one specific space ($C(\Omega)$, $L^1(\mu)$ or $\mathcal{L}^{\infty}(\Gamma)$-type spaces) into the unit sphere of another Banach space. Namely, the isometry owns a linear norm-1 operator as a left-inverse. Finally, we obtain the relationship between the Tingley 问题 and the Figiel type 问题 on unit spheres.