On the isometric extension problem

Abstract

In this paper, we study the isometric extension problem between l ^∞ (Γ, E) and l ^∞ (▵, E), where E is a real uniformly convex Banach space. We obtain that if Γ > 1, then any surjective isometric operator from the unit sphere of l ^∞ (Γ, E) to the unit sphere of l ^∞ (▵, E) can be extended to a linear isometric operator on l ^∞ (Γ, E).