On the quasi-Figiel problem and extension of ε-isometry on unit sphere of L∞,1+ space

This paper aims to study the extension of a bijective epsilon-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(infinity)(n). Based on this quasi-anti-Lipschitz type inequality, the corresponding quasi-anti-Lipschitz type inequality is also obtained for an e-isometric embedding defined on the unit sphere of an L-infinity,L-1+ space, i.e., a Banach space whose finite-dimensional subspaces can be any close to the finite-dimensional subspaces of l(infinity) in the sense of the Banach-Mazur distance. As an application of the quasi-anti-Lipschitz type inequality, we show that every bijective e-isometry between the unit spheres of an L-infinity,L-1+ space and another Banach space can be extended to a bijective 5 epsilon-isometry between their corresponding unit balls. In particular, this implies that every L-infinity,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. (c) 2023 Elsevier Inc. All rights reserved.