Linear isometries between subspaces of continuous functions

We say that a linear subspace A of C-0(X) is strongly separating if given any pair of distinct points x(1), x(2) of the locally compact space X, then there exists f is an element of A such that \f(x(1))\not equal\f(x(2))\. In this paper we prove that a linear isometry T of A onto such a subspace B of C-0(Y) induces a homeomorphism h between two certain singular subspaces of the Shilov boundaries of B and A, sending the Choquet boundary of B onto the Choquet boundary of A. We also provide an example which shows that the above result is no longer true if we do not assume A to be strongly separating. Furthermore we obtain the following multiplicative representation of T: (Tf)(y) = a(y)f(h(y)) for all y is an element of partial derivative B and all f is an element of A, where a isa unimodular scalar-valued continuous function on partial derivative B. These results contain and extend some others by Amir and Arbel, Holsztynski, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.