Lipschitz spaces and M-ideals

Abstract

For a metric space (K,d) the Banach space Lip(K) consists of all scalar-valued bounded Lipschitz functions on K with the norm ||f||_L = max(||f||_{\infty}, L(f)), where L(f) is the Lipschitz constant of f. The closed subspace lip(K) of Lip(K) contains all elements of Lip(K) satisfying the lip-condition lim_{0 < d(x,y) -> 0} |f(x)-f(y)|/d(x,y) = 0. For K=([0,1], |.|^{alpha}), 0 < alpha < 1, we prove that lip(K) is a proper M-ideal in a certain subspace of Lip(K) containing a copy of l^{\infty}.