On properness of uniform algebras

Abstract

In this paper, it is shown that neither proper uniform algebras with closed Choquet boundaries nor proper regular uniform algebras are not isomorphic to C(K)-spaces with respect to the structure of Birkhoff-James orthogonality. This gives a geometric nonlinear version of the fact that proper uniform algebras are not isomorphic to C(K)-spaces. Moreover, some open problems concerning geometric structure spaces of Banach spaces are solved.