The Arens-Calderon theorem forcommutative topological algebras

• M. Weigt ^[2] ; Ioannis Zarakas ^[1]
  1. [1] National and Kapodistrian University of Athens

     National and Kapodistrian University of Athens

     Dimos Athens, Grecia

     
  2. [2] Department of Mathematics, Summerstrand Campus (South) Nelson Mandela University, 6031 Port Elizabeth (Gqeberha), South Africa
• Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 39, Nº 1, 2024, págs. 19-35
• Idioma: inglés
• DOI: 10.17398/2605-5686.39.1.19
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• Resumen

  • A theorem of Arens and Calderon states that if A is a commutative Banach algebra with Jacobson radical Rad(A), and if a0 , . . . , an∈ A with a0 ∈ Rad(A) and a1 an invertible element of k A, then there exists y ∈ Rad(A) such that Σ ak yk = 0. In this paper, we give extensions of this result to commutative non-normed topological algebras, as this is vital for extending an embedding theorem of Allan in [2] regarding the embedding of the formal power series algebra C[[X]] into a commutative Banach algebra

• Referencias bibliográficas
  • M. Abel, Gel’fand-Mazur algebras, in “Topological vector spaces, algebras and related areas” (Hamilton, ON, 1994), Pitman Res. Notes Math....
  • G.R. Allan, Embedding the algebra of formal power series in a Banach algebra, Proc. London Math. Soc. 25 (1972), 329 – 340.
  • G.R. Allan, Fréchet algebras and formal power series, Studia Math. 119 (1996), 271 – 288.
  • R. Arens, The analytic functional calculus in commutative topological algebras, Pacific J. Math. 11 (1961), 405 – 429.
  • R. Arens, A.P. Calderon, Analytic functions of several Banach algebra elements, Ann. of Math. (2) 62 (1955), 204 – 216.
  • F.F. Bonsall, J. Duncan, “ Complete normed algebras ”, Springer-Verlag, New York-Heidelberg, 1973.
  • H. Biller, Analyticity and naturality of the multi-variable functional calculus, Expo. Math. 25 (2007), 131 – 163.
  • A.V. Ferreira, G. Tomassini, Finiteness properties of topological algebras, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 471 – 488.
  • M. Fragoulopoulou, “Topological algebras with involution”, North-Holland Math. Stud., 200, Elsevier Science B.V., Amsterdam, 2005.
  • M. Fragoulopoulou, A. Inoue, M. Weigt, I. Zarakas, “Generalized B∗ -algebras and applications”, Lecture Notes in Math., 2298, Springer, Cham,...
  • L. Hormander, “An introduction to complex analysis in several variables”, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London,...
  • J. Lindenstrauss, L. Tzafriri, “Classical Banach spaces. I. Sequence spaces”, Ergeb. Math. Grenzgeb., Band 92, Springer-Verlag, Berlin-New...
  • H.H. Schaefer, “Topological vector spaces”, Grad. Texts in Math., Vol. 3, Springer-Verlag, New York-Berlin, 1971.
  • L. Waelbroeck, “Topological vector spaces and algebras”, Lecture Notes in Math., Vol. 230, Springer-Verlag, Berlin-New York, 1971.
  • L. Waelbroeck, The holomorphic functional calculus as an operational calculus, in “Spectral theory”, Banach Center Publ., 8, PWN–Polish Scientific...
  • M. Weigt, I. Zarakas, On formal power series over topological algebras, Extracta Math. 37 (2022), 57 – 74.
  • W. Zelazko, “Metric generalizations of Banach algebras”, Rozprawy Mat. 47 (1965), 70 pp.