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Isometries and the complex state spaces of uniform algebras. (English) Zbl 0599.46074

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.

MSC:

46J10 Banach algebras of continuous functions, function algebras
46A55 Convex sets in topological linear spaces; Choquet theory
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References:

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