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Countable inductive limits of Fréchet algebras. (English) Zbl 0905.46032
Summary: We show that the Gelfand-Mazur theorem holds for countable inductive limits of Fréchet algebras (we do not assume that the homomorphisms which define the inductive limit are continuous, or one-to-one). This question is motivated by the fact that the spectrum of some elements of such an algebra may be empty. We also discuss in detail a countable inductive limit of Fréchet algebras of holomorphic functions, which provides an elementary, but seminal, counterexample to the biinvariant subspace problem for complete, reflexive, locally convex spaces.

MSC:
46H05 General theory of topological algebras
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A04 Locally convex Fréchet spaces and (DF)-spaces
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