On the quasi-Figiel problem and extension of ε-isometry on unit sphere of [formula omitted] space

Abstract

This paper aims to study the extension of a bijective ε-isometry and the existence of the Figiel operator of an isometric embedding between two unit spheres of Banach spaces. In the first part, we introduce and study the quasi-Figiel problem about a quasi-isometric embedding between unit spheres of two Banach spaces. Consequently, a quasi-anti-Lipschitz type inequality is obtained when the domain space is l∞n. Based on this quasi-anti-Lipschitz type inequality, the corresponding quasi-anti-Lipschitz type inequality is also obtained for an ε-isometric embedding defined on the unit sphere of an L∞,1+ space, i.e., a Banach space whose finite-dimensional subspaces can be any close to the finite-dimensional subspaces of l∞ in the sense of the Banach-Mazur distance. As an application of the quasi-anti-Lipschitz type inequality, we show that every bijective ε-isometry between the unit spheres of an L∞,1+ space and another Banach space can be extended to a bijective 5ε-isometry between their corresponding unit balls. In particular, this implies that every L∞,1+ space admits the Mazur-Ulam property. Furthermore, in this paper, some attempts are also made to generalize the classical Figiel's theorem to the local case with respect to an isometric embedding between unit spheres.