2-Local Isometries on Spaces of Lipschitz Functions

Let (X, d) be a metric space, and let Lip(X) denote the Banach space of all scalar-valued bounded Lipschitz functions f on X endowed with one of the natural norms parallel to f parallel to = max{parallel to f parallel to(infinity), L(f)} or parallel to f parallel to = parallel to f parallel to(infinity) + L(f), where L(f) is the Lipschitz constant of f. It is said that the isometry group of Lip(X) is canonical if every surjective linear isometry of Lip(X) is induced by a surjective isometry of X. In this paper we prove that if X is bounded separable and the isometry group of Lip(X) is canonical, then every 2-local isometry of Lip(X) is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of Lip(X) when X is bounded.