Representation in C (K) by Lipschitz functions

The isometric universality of the spaces C (K) for K a non scattered Hausdorff compact does not take into account the ``quality'' of the representation. Indeed, the existence of an isometric copy of a separable Banach space X into C (K) made of regular enough functions, say Lipschitz with respect to a lower semicontinuous metric defined on K, imposes severe restrictions to both X and K. In this paper, we present a systematic treatment of the representation of Banach spaces into C (K) by Lipschitz functions improving previous results of the author.