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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.

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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