Automatic continuity, bases, and radicals in metrizable algegbras
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The automatic continuity of a linear multiplicative operator T: X→Y, where X and Y are real complete metrizable algebras and Y semi-simple, is proved. It is shown that a complex Frechét algebra with absolute orthogonal basis (xi) (orthogonal in the sense that xiXj=0 if i ≠ j) is a commutative symmetric involution algebra. Hence, we are able to derive the well-known result that every multiplicative linear functional defined on such an algebra is continuous. The concept of an orthogonal Markushevich basis in a topological algebra is introduced and is applied to show that, given an arbitrary closed subspace Y of a separable Banach space X, a commutative multiplicative operation whose radical is Y may be introduced on X. A theorem demonstrating the automatic continuity of positive functionals is proved.
KeywordsBanach Space Closed Subspace Orthogonal Basis Separable Banach Space Topological Algebra
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- 1.T. Husain, Multiplicative Functionals on Topological Algebras, Research Notes in Math., Vol. 85, Pitman, Boston (1983).Google Scholar
- 2.T. Husain and Shu-Bun Ng, “On continuity of algebra homomorphisms and uniqueness of metric topology,” Math. Z.,139, 1–4 (1974).Google Scholar
- 3.Yu. I. Petunin and V. D. Pogrebnoi, “Certain problems of embedding of quotient spaces in Banach algebras,” Ukr. Mat. Zh.,37, No. 1, 87–93 (1985).Google Scholar
- 4.F. Gregory and S. Saeki, “Banach algebras with uncomplemented radical,” Proc. Amer. Math. Soc.,100, No. 2, 271–274 (1987).Google Scholar
- 5.A. Pelczynski, “All separable Banach spaces admit for every ɛ > 0 fundamental total and bounded by 1 + ɛ biorthogonal sequences,” Stud. Math.,55, No. 3, 295–304 (1976).Google Scholar
- 6.M. I. Kadets and B. S. Mityagin, “Complementable subspaces in Banach spaces,” Usp. Mat. Nauk,28, No. 6, 77–94 (1973).Google Scholar
- 7.A. M. Sinclair, “Automatic continuity of linear operators,” London Math. Soc. Lect. Notes,21, 1–92 (1976).Google Scholar
- 8.P. G. Dixon, “Automatic continuity of positive functionals on topological involution algebras,” Bull. Austral. Math. Soc.,23, No. 2, 265–281 (1981).Google Scholar
- 9.G. Godefroy, “Certain properties of Banach spaces,” Semin. Choquet initiat, anal. Univ. Pierre et Marie Curie,14, C3/1-C3/8 (1974–1975).Google Scholar
- 10.I. V. Yakovlev, “Examples of Banach algebras with radical noncomplementable as a Banach space,” Usp. Mat. Nauk,44, No. 5, 185–186 (1989).Google Scholar