×

Continuité des caractères dans les algèbres de Fréchet à bases. (Continuity of characters in Fréchet algebras with basis). (French) Zbl 0663.46046

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

MSC:

46H15 Representations of topological algebras
46H20 Structure, classification of topological algebras
46H05 General theory of topological algebras

Citations:

Zbl 0365.46038
PDFBibTeX XMLCite
Full Text: DOI