Extreme points and isometries on vector-valued Lipschitz spaces

Abstract

For a Banach space E and a compact metric space ( X , d ) , a function F : X → E is a Lipschitz function if there exists k > 0 such that ‖ F ( x ) − F ( y ) ‖ ⩽ k d ( x , y ) for all x , y ∈ X . The smallest such k is called the Lipschitz constant L ( F ) for F. The space Lip ( X , E ) of all Lipschitz functions from X to E is a Banach space under the norm defined by ‖ F ‖ = max { L ( F ) , ‖ F ‖ ∞ } , where ‖ F ‖ ∞ = sup { ‖ F ( x ) ‖ : x ∈ X } . Recent results characterizing isometries on these vector-valued Lipschitz spaces require the Banach space E to be strictly convex. We investigate the nature of the extreme points of the dual ball for Lip ( X , E ) and use the information to describe the surjective isometries on Lip ( X , E ) under certain conditions on E, where E is not assumed to be strictly convex. We make use of an embedding of Lip ( X , E ) into a space of continuous vector-valued functions on a certain compact set.