Linear isometries between subspaces of continuous functions

Abstract

We say that a linear subspace A A of C 0 ( X ) C_0 (X) is strongly separating if given any pair of distinct points x 1 , x 2 x_1, x_2 of the locally compact space X X , then there exists f ∈ A f \in A such that | f ( x 1 ) | ≠ | f ( x 2 ) | \left | f(x_1 ) \right | \neq \left | f(x_2 ) \right | . In this paper we prove that a linear isometry T T of A A onto such a subspace B B of C 0 ( Y ) C_0(Y) induces a homeomorphism h h between two certain singular subspaces of the Shilov boundaries of B B and A A , sending the Choquet boundary of B B onto the Choquet boundary of A A . We also provide an example which shows that the above result is no longer true if we do not assume A A to be strongly separating. Furthermore we obtain the following multiplicative representation of T T : ( T f ) ( y ) = a ( y ) f ( h ( y ) ) (Tf)(y)=a(y)f(h(y)) for all y ∈ ∂ B y \in \partial B and all f ∈ A f \in A , where a a is a unimodular scalar-valued continuous function on ∂ B \partial B . These results contain and extend some others by Amir and Arbel, Holsztyński, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.