Min-Phase-Isometries on the Unit Sphere of L¹ Spaces

Let (Ω , M, μ ) be a measure space. We show that if a surjective map f: SL¹(μ ) → SY between the unit spheres of real L¹(μ ) and of an arbitrary real normed space Y satisfies align* min\\|f(x)+f(y)\|, \|f(x)-f(y)\|\=min\\|x+y\|, \|x-y\|\, x,y∈ SL¹(μ), align* then there exists a phase function ε : SL¹(μ ) → \-1, 1\ such that ε · f is a surjective isometry from SL¹(μ ) onto SY , and furthermore, this isometry can be extended to a linear isometry on the whole space L¹(μ ) .