On extension of isometries between unit spheres of L∞(Γ)-type space and a Banach space E

In this paper, we study the isometric extension problem between the unit spheres of L∞(Γ)-type space and a Banach space E. We prove that every surjective isometry between the unit spheres S(L∞(Γ)) and S(E) can be extended to be a linear isometry of L∞(Γ) onto E. Moreover, we study this problem between C(Ω) and a Banach space and also get an affirmative answer.