Topological algebras with finitely-generated bases (Q797809)

We study the structure of algebras with Schauder bases generated multiplicatively by a finite set of elements \(z_ 1,...,z_ n\). We show that the algebra is semisimple if and only if the joint spectrum \(\sigma (z_ 1,...,z_ n)\) has nonempty interior and that when \(\sigma (z_ 1,...,z_ n)\) is open it is homeomorphic to the maximal ideal space. Algebras for which \(\sigma (z_ 1,...,z_ n)\) is open are characterized as algebras of holomorphic functions on Reinhardt domains.