The Arens-Calderon theorem for commutative topological algebras


  • M. Weigt Department of Mathematics, Summerstrand Campus (South) Nelson Mandela University, 6031 Port Elizabeth (Gqeberha), South Africa
  • I. Zarakas Department of Mathematics, University of Athens Panepistimiopolis, Athens 15784, Greece



Formal power series, theorem of Arens and Calderon, commutative topological algebra, functional calculus


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.


Download data is not yet available.


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.






Functional Analysis and Operator Theory

How to Cite

The Arens-Calderon theorem for commutative topological algebras. (2024). Extracta Mathematicae, 39(1), 19-35.

Most read articles by the same author(s)