Boundedness Of Involutions In Topological Algebras With Orthonormal Generalized Bases

Abstract

A main result of this article establishes that every involution in a sequentially complete locally convex topological algebra with an orthonormal generalized basis is bounded. This result is obtained through a representation for involutions in these algebras.

Keywords

• Net convergence
• Automatic continuity
• Schauder bases
• Topological algebras.

References

1. M. E. Azhari, Functional continuity of unital B0-algebras with orthogonal bases, Le matematiche, 72 (2017), 97-102
2. V.K. Balachandran, Topological Algebras, North-Holland, Amsterdam, 2000.
3. S. J. Bhatt and G. M. Deheri, Kothe spaces and topological algebra with bases, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 3, 1990, pp. 259-273
4. S. J. Bhatt and G. M. Deheri, Orthogonal bases in a topological algebra are Schauder bases, Internat. J. Math. & Math. Sci. 15 (1992), 203-204
5. P.G. Dixon and D.H. Fremlin, A remark concerning multiplicative functionals on LMC algebras, J. Lond. Math. Soc. 5 (1972) 231–232
6. T. Husain, Positive functionals on topological algebras with bases, Math. Japonica, 28 (1983), 683-687.
7. T. Husain, J. Liang, Multiplicative functionals on Frechet algebras with bases, Canad. J. Math. 29 (1977), 270-276
8. T. Husain, J. Liang, Continuity of multiplicative linear functionals on Frechet algebras with bases, Bull. Soc. Roy. Sc Li´ege, 46 (1977), 8-11

9. Husain T and Watson S, Topological algebras with orthogonal Schauder basis, Pacific J. Math. 91 (1980) 339-347
10. Husain T and Watson S, Algebras with unconditional orthogonal basis, Proc. Am. Math. Soc. 79 (1980) 539-545
11. S. Loganathan and C.G. Moorthy, A net convergence for Schauder double bases, Asian-Eur. J. Math. 9 (2016) 33 pp.
12. E.A. Michael, Locally Multiplicatively Convex Topological Algebras, Mem. Amer. Math. Soc. 11, Amer. Math. Soc. Providence, RI, 1952.
13. AR. Murugan, C. Ganesa Moorthy and CT. Ramasamy, Directed bases with net convergence, Surveys in Mathematics and its Applications, 19 (2024), 57 – 66.
14. W. Rudin, Functional Analysis, Mc Graw Hill, New York, 1973.
15. Z. Sawon and Z. Wronski, Fr´echet algebras with orthogonal bases, Colloq. Math. 48 (1984), 103-110
16. I. Singer, Bases in Banach Spaces I, Springer, New York, 1970.
17. G. Siva and C.G. Moorthy, Functional continuity of topological algebras with orthonormal bases, Asian-Eur. J. Math. 14 (2021) Article no. 2150059, 15 pp
18. G. Siva, and C.G. Moorthy, Reviewed techniques in automatic continuity of linear functionals, Khayyam J. Math. 9 (2023), no. 1, 1-29