Continuity of homomorphisms on complete metrizable topological algebras

Document Type : Research Paper


Department of Mathematics, Alagappa University, Karaikudi-630 003, India


Let $A$ be an $F$-algebra over the complex field.  Let $B$ be an $F$-algebra such that the intersection of kernels of all continuous multiplicative linear functionals on $B$ is singleton zero.  If any nonempty open subset of the collection $M(B)$ of all continuous multiplicative linear functionals in the Gelfand topology contains uncountably many functionals or if $B$ is a commutative Frechet algebra such that $M(B)$ has no isolated points, then any homomorphism from $A$ onto a dense finitely generated subalgebra of $B$ is continuous.  This result has been proved in this article which is similar to a result derived by R.L. Carpenter.


[1] M. E. Azhari, Functional continuity of unital B0-algebras with orthogonal bases, Le Matematiche 72 (2017),
[2] R. L. Carpenter, Uniqueness of topology for commutative semisimple F-algebras, Proc. Amer. Math. Soc. 29
(1971), 113–117.
[3] S.M. Corson and I. Kazachkov, On preservation of automatic continuity, Arxiv: 1901.0901.09279v1 [math.GR],
(2019), 1-14.[4] P.A. Dabhi and H.V. Dedania, On the uniqueness of uniform norms and C

-norms, Studia Math. 191 (2009),
[5] A.P. Farajzadeh and M.R. Omidi, Almost multiplicative maps and ε-spectrum of an element in Fr´echet Q-algebra,
Filomat 33 (2019), 1445–1452.
[6] M. Eshaghi Gordji, A. Jabbari and E. Karapinar, Automatic continuity of n-homomorphisms between Banach
algebras, Bull. Iran. Math. Soc. 41 (2015), 1207–1211.
[7] T. G. Honary, Automatic continuity of homomorphisms between Banach algebras and Frechet algebras, Bull.
Iranian Math. Soc., 32 (2006), 1-11.
[8] T.G. Honary, M. Omidi and A.H. Sanatpour, Automatic continuity of almost multiplicative linear functionals on
Fr´echet algebras, Bull. Korean Math. Soc. 53 (2016), 641–649.
[9] E. A. Michael, Locally multiplicatively convex topological algebras, Mem. Amer. Math. Soc., 11 (1952).
[10] C.G. Moorthy and G. Siva, Automatic continuity of Jordan almost multiplicative maps on special Jordan Banach
algebras, Eur. J. Math. Appl. 2 (2022), 1–6.
[11] M.R. Omidi, A.P. Farajzadeh, E. Soori and B.O. Gillan, Automatic continuity on fundamental locally multiplicative topological algebras, Thai J. Math. 17 (2019), 155–164.
[12] I. Pastukhova, Automatic continuity of homomorphisms between topological inverse semigroups, Topol. Algebra
Appl. 6 (2019), 60–66.
[13] S.R. Patel, Uniqueness of the Frechet algebra topology on certain Frechet algebras, Studia Math. 234 (2016), 31–47.
[14] M. Rosenfeld, Commutative F-algebras, Pacific J. Math. 16 (1966), 159–166.
[15] G. Siva and C.G. Moorthy, Uniqueness of F-algebra topology for commutative semisimple algebras, Bull. Iran.
Math. Soc. 45 (2019), 1871–1877.
[16] G. Siva and C.G. Moorthy, Functional continuity of topological algebras with orthonormal bases, Asian-Eur. J.
Math. 14 (2020), 1–15.
[17] R. Skillicorn, The uniqueness-of-norm problem for Calkin algebras, Math. Proc. of Royal Irish Acad. 2 (2015),
[18] A. R. Villena, Uniqueness of the topology on spaces of vector-valued functions, J. London Math. Soc. 64 (2001),
[19] A. Zivari-Kazempour, Automatic continuity of n-Jordan homomorphisms on Banach algebras, Commun. Korean
Math. Soc. 33 (2018), 165–170.
Volume 13, Issue 2
July 2022
Pages 1983-1987
  • Receive Date: 18 December 2020
  • Revise Date: 08 March 2021
  • Accept Date: 28 May 2021