The phase-isometries between the unit sphere of lp(Γ,H)-type spaces

Abstract

In this paper, Suppose that f: SX → SY is a surjective phase-isometry between the unit spheres of two real l_p(Γ,H)-type spaces X and Y. We prove that the mapping f is phase equivalent to an isometry. Otherwise, this isometry is the restriction of a linear isometry between the whole spaces, i.e., this isometry on the unit sphere can be linearly extended into isometry in the whole space.