A note on extension of isometric embedding from a Banach space E into the universal space

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.