Akkar, M.; El Azhari, M.; Oudadess, M. Continuité des caractères dans les algèbres de Fréchet à bases. (Continuity of characters in Fréchet algebras with basis). (French) Zbl 0663.46046 Can. Math. Bull. 31, No. 2, 168-181 (1988). Let A be a complete locally m-convex algebra. In this note the authors give short proofs of the following theorems:1. If A is commutative with a Schauder basis \((x_ i)_{i\geq 1}\) satisfying \(x_ ix_ j=x_ j\), \(i\leq j\), then A is semi-simple; if, moreover, the series \(\sum^{\infty}_{i=1}x_ i\) converges, then all characters of A are continuous. 2. If A is a Fréchet algebra with an orthogonal basis \((x_ i)_{i\geq 1}\) such that \(\sum^{\infty}_{i=1}a_ ix_ i\in A\) for every sequence \((a_ i)_{i\geq 1}\) of complex numbers, then each character of A is continuous. These theorems are improvements of some results obtained by T. Husain and J. Liang [Bull. Soc. Roy. Sc. Liège 46, 8-11 (1977; Zbl 0365.46038)]. Reviewer: I.Vidav Cited in 1 Document MSC: 46H15 Representations of topological algebras 46H20 Structure, classification of topological algebras 46H05 General theory of topological algebras Keywords:character; locally m-convex algebra; Schauder basis; Fréchet algebra with an orthogonal basis Citations:Zbl 0365.46038 PDFBibTeX XMLCite \textit{M. Akkar} et al., Can. Math. Bull. 31, No. 2, 168--181 (1988; Zbl 0663.46046) Full Text: DOI