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Fréchet algebras with orthogonal basis. (English) Zbl 0569.46026

A basis \(\{e_ n\}\) in a topological algebra is called orthogonal if \(e_ ne_ m=\delta_{nm}e_ n\)- a concept introduced by this reviewer and studied in collaboration with others. In their paper [Can. J. Math. 29, 270-276 (1977; Zbl 0348.46036)], J. Liang and this reviewer show that each multiplicative linear functional on a Fréchet (complete metrizable locally m-convex) algebra with an unconditional orthogonal basis is continuous. Here the authors show that the same is true on a \(B_ 0\)-algebra (complete metrizable locally convex) A with unit and orthogonal basis, provided there exists a monotonically increasing sequence \(\{\lambda_ i\}\) of real numbers, \(\lim_{n} \lambda_ n=+\infty\) such that \(\sum \lambda_ ne_ n\in A\). Among other results, some conditions are given under which a \(B_ 0\)-algebra with an orthogonal basis and identity is isomorphic with the Fréchet algebra of all complex sequences, thus giving an alternative proof of a result in the above-mentioned paper. [For other relevant results see T. Husain and S. Watson, Pac. J. Math. 91, 339-347 (1980; Zbl 0477.46042); Proc. Am. Math. Soc. 79, 539-545 (1980; Zbl 0434.46029)].
Reviewer: T.Husain

MSC:

46H20 Structure, classification of topological algebras
46H05 General theory of topological algebras
46A35 Summability and bases in topological vector spaces
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