LINEAR ISOMETRIES BETWEEN SUBSPACES OF CONTINUOUS-FUNCTIONS

We say that a linear subspace A of C-0(X) is strongly separating if gi ven any pair of distinct points x(1), x(2) of the locally compact spac e X, then there exists f is an element of A such that \f(x(1))\not equ al\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 cer tain singular subspaces of the Shilov boundaries of B and A, sending t he Choquet boundary of B onto the Choquet boundary of A. We also provi de an example which shows that the above result is no longer true if w e do not assume A to be strongly separating. Furthermore we obtain the following multiplicative representation of T: (Tf)(y) = a(y)f(h(y)) f or 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 par tial derivative B. These results contain and extend some others by Ami r and Arbel, Holsztynski, Myers and Novinger. Some applications to iso metries involving commutative Banach algebras without unit are announc ed.