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On Fréchet algebras of power series

  • S. J. Bhatt (a1) and S. R. Patel (a1)

Abstract

If the indeterminate X in a Fréchet algebra A of power series is a power series generator for A, then either A is the algebra of all formal power series or is the Beurling-Fréchet algebra on non-negative integers defined by a sequence of weights. Let the topology of A be defined by a sequence of norms. Then A is an inverse limit of a sequence of Banach algebras of power series if and only if each norm in the defining sequence satisfies certain closability condition and an equicontinuity condition due to Loy. A non-Banach uniform Fréchet algebra with a power series generator is a nuclear space. A number of examples are discussed; and a functional analytic description of the holomorphic function algebra on a simply connected planar domain is obtained.

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References

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[1]Allan, G.R., ‘Commutative Banach algebras with a power series generator’, in Radical Banach algebras and automatic continuity, (Bachar, J.M., Bade, W.G., Curtis, P.C. Jr., Dales, H.G. and Thomas, M.P., Editors), Lecture Notes in Mathematics 975 (Springer-Verlag, Berlin. 1983), pp. 290294.
[2]Allan, G.R., ‘Fréchet algebras and formal power series’, Studia Math. 119 (1996), 271288.
[3]Carboni, G. and Larotonda, A., ‘An example of a Fréchet algebra which is a principle ideal domain’, Studia Math. 138 (2000), 265275.
[4]Gelfand, I.M., Raikov, D. and Silov, G., Commutative normed rings (Chelsea Publ. Co., Bronx, New York, 1964).
[5]Goldmann, H., Uniform Fréchet algebras, North Holland Mathematical Studies 162 (North Holland Publ. Co., Amsterdam, 1990).
[6]Husain, T., Orthogonal Schauder basis, Monographs and Textbooks in Pure and Applied Mathematics 134 (Marcel-Dekker Inc., New York, 1991).
[7]Loy, R.J., ‘Continuity of derivations on topological algebras of power series’, Bull. Austral. Math. Soc. 1 (1969), 419424.
[8]Loy, R.J., ‘Uniqueness of the Fréchet space topology on certain topological algebras’, Bull. Austral. Math. Soc. 4 (1971), 17.
[9]Loy, R.J., ‘Banach algebras of power series’, J. Austral. Math. Soc. 17 (1974), 263273.
[10]Pietsch, A., Nuclear locally convex spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1972).
[11]Zelazko, W., ‘On maximal ideals in commutative m-convex algebras’, Studia Math. 58 (1976), 291298.
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