A note on extension of isometric embedding from a Banach space E into the universal space ℓ∞(Γ)

In this paper we shall present a short proof on the extension problem of isometric embedding between unit spheres of a Banach space E and the universal space ℓ∞(Γ). We prove that, under some condition, every isometric embedding from S(E) into S(ℓ∞(Γ)) can be positive-homogeneously isometrically extended to the whole space. Since every Banach space E is isometric to a subspace of ℓ∞(S(E∗)), isometric extension problems on a class of atomic AM-spaces is solved.