On Isometric Extension in the Spacesn(H)

Abstract

We study the problem of isometric extension on a sphere of the spacesn(H). We give an affirmative answer to Tingley’s problem in the spacesn(H).

Tingley's problem on uniform algebras

“…He dealt with C 0 (Ω), the space of all complex-valued continuous functions on a locally compact Hausdorff space Ω which vanish at infinity. A considerable number of interesting results have shown that Tingley's problem has a positive solution in ℓ p spaces, L p spaces, C 0 (Ω) spaces, certain finite dimensional Banach spaces, and so on [7,8,10,11,13,15,16,26,27,28,29,30,34,53,54,56,58,59,62,67,68,69,70,71,72,74,76].…”

The problem of isometric extension on the unit sphere of the space $ l\cap l^p(H)$ for $0 < p < 1$

“…Kadets and Martin in [6] proved that any surjective isometry between unit spheres of finite-dimensional polyhedral Banach spaces has a linear isometric extension to the whole space. See also [2,3,4,5,9,11,12] for some related results.…”