Maximal invariant subalgebras in algebras with involution

A very far-reaching generalization of Wermer's well-known theorem on the maximality of the subalgebra of analytic functions has been obtained in recent papers by Bonsall and Ford. The algebra of analytic functions of several variables, on the contrary, is not maximal. However, the standard algebra of analytic functions on the skeleton of a polycylinder is maximal among the subalgebras invariant under the natural semigroup of analytic endomorphisms. A general scheme is proposed in which this example occupies the same place as Wermer's theorem occupies in Bonsall's scheme.
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