The Arens-Calderon theorem for commutative topological algebras
DOI:
https://doi.org/10.17398/2605-5686.39.1.19Palabras clave:
Formal power series, theorem of Arens and Calderon, commutative topological algebra, functional calculusResumen
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.
Descargas
Referencias
M. Abel, Gel’fand-Mazur algebras, in “Topological vector spaces, algebras and related areas” (Hamilton, ON, 1994), Pitman Res. Notes Math. Ser. 316, Longman Sci. Tech., Harlow, 1994, 116 – 129.
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, 2022.
L. Hormander, “An introduction to complex analysis in several variables”, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966.
J. Lindenstrauss, L. Tzafriri, “Classical Banach spaces. I. Sequence spaces”, Ergeb. Math. Grenzgeb., Band 92, Springer-Verlag, Berlin-New York, 1977.
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 Publishers, Warsaw, 1982, 513 – 552.
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.
Descargas
Publicado
Número
Sección
Licencia
Derechos de autor 2024 The authors
Esta obra está bajo una licencia internacional Creative Commons Atribución-NoComercial 4.0.
