Isometries and the complex state spaces of uniform algebras (Q1080125)

If \({\mathfrak A}\) is a uniform algebra with state space \(S=\{\phi \in {\mathfrak A}^*:\phi (\ell)=\ell =\| \phi \| \}\) then the \(w^*\)-compact convex subset \(Z=Co(S\cup -iS)\) of \({\mathfrak A}^*\) is called the complex state space of \({\mathfrak A}\). It is shown that S and Z are determined by the isometries of A(S) and A(Z) respectively, and we examine the relationship between (complex-linear) isometries of \({\mathfrak A}\) and (real-linear) isometries of A(Z). The results make substantial use of the facial topology for S and Z. When the maximal ideal space of \({\mathfrak A}\) has only finitely many connected components it is shown that the affine geometry of Z determines the algebraic structure of \({\mathfrak A}\) up to complex conjugation.