Topological algebras with subadditive boundedness radius

  • M. Sabet Department of Mathematics, Payame Noor University, Tehran, Iran.
  • R. G. Sanati Institute of Higher Education of ACECR (Academic Center of Education and Culture Research), Rasht branch, Iran.
Keywords: Topological algebra, strongly sequential algebra, boundedness radius


Let \(A\) be a topological algebra and \(\beta\) a subadditive boundedness radius on \(A\). In this paper we show that \(\beta\) is, under certain conditions, automatically submultiplicative. Then we apply this fact to prove that the spectrum of any element of \(A\) is non-empty. Finally, in the case when \(A\) is a normed algebra, we compare the initial normed topology with the normed topology \(\tau_{\beta}\), induced by \(\beta\) on \(A\), where \(\beta^{-1} (0)=0\).


G. R. Allan, A spectral theory for locally convex algebras, Proc. London Math. Soc. (3) vol. 15, pp. 399–421, 1965.

T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo, vol. 18, pp. 588– 594, 1942.

E. Ansari-Piri, M. Sabet and S. Sharifi, A class of complete metrizable Q-algebras, Sci. Stud. Res. Ser. Math. Inform. vol. 26, no. 1, pp. 17–24, 2016.

S. J. Bhatt, A seminorm with square property on a Banach algebra is submultiplicative, Proc. Amer. Math. Soc. vol. 117, no. 2, pp. 435–438, 1993.

F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, New York, 1973.

H. V. Dedania, A seminorm with square property is automatically submultiplicative, Proc. Indian Acad. Sci. Math. Sci. vol. 108 1998, no. 1, pp. 51–53, 1998.

T. Husain, Multiplicative functionals on topological algebras, Research Notes in Mathematics, 85, Pitman (Advanced Publishing Program), Boston, MA, 1983.

A. El Kinani, L. Oubbi and M. Oudadess, Spectral and boundedness radii in locally convex algebras, Georgian Math. J. vol. 5, no. 3, pp. 233–241, 1998.

A. Mallios, Topological algebras. Selected topics, North-Holland Mathematics Studies, 124, North-Holland Publishing Co., Amsterdam, 1986.

G. J. Murphy, C∗-algebras and operator theory, North-Holland. 1990.

L. Oubbi, Further radii in topological algebras, Bull. Belg. Math. Soc. Simon Stevin vol. 9, no. 2, pp. 279–292, 2002.

W. Zélazko, On the locally bounded and m-convex topological algebras, Studia Math. vol. 19, pp. 333–356, 1960.

How to Cite
M. Sabet and R. G. Sanati, “Topological algebras with subadditive boundedness radius”, CUBO, vol. 22, no. 3, pp. 289–297, Dec. 2020.