Abstract
We say that a linear subspaceAAofC0(X)C_0 (X)is strongly separating if given any pair of distinct pointsx1,x2x_1, x_2of the locally compact spaceXX, then there existsf∈Af \in Asuch that|f(x1)|≠|f(x2)|\left | f(x_1 ) \right | \neq \left | f(x_2 ) \right |. In this paper we prove that a linear isometryTTofAAonto such a subspaceBBofC0(Y)C_0(Y)induces a homeomorphismhhbetween two certain singular subspaces of the Shilov boundaries ofBBandAA, sending the Choquet boundary ofBBonto the Choquet boundary ofAA. We also provide an example which shows that the above result is no longer true if we do not assumeAAto be strongly separating. Furthermore we obtain the following multiplicative representation ofTT:(Tf)(y)=a(y)f(h(y))(Tf)(y)=a(y)f(h(y))for ally∈∂By \in \partial Band allf∈Af \in A, whereaais 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.
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