Abstract
Let A be a radical Banach algebra and \(A'\) its topological dual. We show that the multiplication in A is weakly continuous if, and only if, A, endowed with its weak topology \(\sigma (A, A')\), is isomorphic to a subalgebra of a product of finite dimensional radical algebras. This answers an open question in the literature. We also show that such an algebra is either finite dimensional or possesses uncountably many closed ideals of finite codimension. We then obtain that a radical Banach algebra with weakly continuous multiplication admits at least one maximal closed ideal. We thus provide a further class of radical Banach algebras, where the answer to the long standing problem of existence of closed ideals is in the affirmative, as for compact ones.
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References
Akkar, M., Albrecht E., Oubbi, L.: A further characterization of finite dimensional Banach algebras. Preprint (1997), available at https://www.researchgate.net/profile/Lahbib-Oubbi
Akkar, M., Oubbi, L., Oudadess, M.: Some results on weakly topologized algebras. J. Univ. Kuwait (Sci.) 19, 1–8 (1992)
Bonsall, F.F., Duncan, J.: Complete Normed Algebras. Springer-Verlag, Berlin (1973)
Chilana, A.K.: Topological algebras with a given dual. Proc. Amer. Math. Soc. 42, 192–197 (1974)
Chilana, A.K.: Hypocontinuity multiplication in weak topologized algebras. Yokohama Math. J. 24(1–2), 97–102 (1976)
Cochran, A.C.: Weak A-convex algebras. Proc. Amer. Math. Soc. 26, 73–77 (1970)
Kaplansky, I.: Ring isomorphisms of Banach algebras. Can. J. Math. 6, 374–381 (1954). https://doi.org/10.4153/CJM-1954-036-5
Michael, E.A.: Locally multiplicatively convex topological algebras. Memoirs American Mathematical Society, vol. 11, (1952)
Oubbi, L.: M-convexity and a-convexity of polar algebra topologies. Math. Res. Appl. 5, 85–95 (2003)
Oubbi, L.: Weak topological algebras and P-algebra property. In: Proceedings of ICTAA 2008, Math. Stud. series 4, Estonian Math. Soc. pp. 73–79, (2008)
Palmer, T. W., Banach algebras and the general theory of *-algebras. Volume 1: Algebras and Banach Algebras, Cambridge University Press (1994). https://doi.org/10.1017/CBO9781107325777
Rickart, C.E.: General Theory of Banach Algebras. Van Nostrand Company, Princeton (1960)
Turovskii, Yu.V., Shulman, V.S.: Radicals in Banach Algebras and some problems in the theory of radical algebras. Funct. Anal. Appl. 35(4), 312–314 (2001). https://doi.org/10.1023/A:1013186826199
Tylli, H.O., Wirzenius, H.: Closed ideals in the algebra of compact-by-approximable operators. J. Funct. Anal. 282(4), 109328 (2022)
Warner, S.: Weakly topologized algebras. Proc. Amer. Math. Soc. 8, 314–316 (1957)
Warner, S.: Weak locally multiplicatively convex algebras. Pac. J. Math. 5, 1025–1032 (1955)
Wojtynski, W.: On the existence of closed two-sided ideals in radical Banach algebras with compact elements. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 26(2), 109–113 (1978)
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Oubbi, L. Radical Banach algebras with weakly continuous multiplication. Rend. Circ. Mat. Palermo, II. Ser 73, 1231–1240 (2024). https://doi.org/10.1007/s12215-023-00980-7
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DOI: https://doi.org/10.1007/s12215-023-00980-7