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Continuité des Caractères Dans les Algèbres de Fréchet à Bases

Published online by Cambridge University Press:  20 November 2018

M. Akkar ,
M. El Azhari  and
M. Oudadess
M. Akkar
Affiliation:
École Normale Supérieure, Avenue Oued AkreuchTakaddoum, RabatB.P. 5118 Maroc
M. El Azhari
Affiliation:
École Normale Supérieure, Avenue Oued AkreuchTakaddoum, RabatB.P. 5118 Maroc
M. Oudadess
Affiliation:
École Normale Supérieure, Avenue Oued AkreuchTakaddoum, RabatB.P. 5118 Maroc
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Abstract

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In [1] and [2] T. Husain and J. Liang show the following results: RI. Every character on a Fréchet algebra with a Schauder basis (xi)i≧1 such that: (1) XiXj = XjXi = Xj if i ≦ j, (2) Pi(Xi) ≠ 0 and Pi(xi+1) = 0 (where (pi)i≧1 is a denumerable family of semi-norms defining the topology of the algebra) is continuous. R2. Every character on a Fréchet algebra with orthogonal and unconditional Schauder basis is continuous. The proofs of these last results are very long and introduce complex calculation without aid of spectral theory of locally ra-convex algebras. We give here short proofs of these results with aid of a characterization of elements of the spectrum in locally m-convex algebras with values of characters.

Keywords

46H15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 2 , 01 June 1988 , pp. 168 - 174
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Husain, T. and Liang, J., Multiplicative functionals on Fréchet algebras with bases, Can. J. Math. Vol XXIX n° 2 (1977), pp. 270–276.Google Scholar
2. Husain, T. and Liang, J., Continuity of multiplicative linear functionals on Fréchet algebras with bases, Bull. Soc. Roy. Se Liège 46 (1977), pp. 8–11.Google Scholar
3. Michaël, E. A., Locally multiplicatively convex topological algebras, Mem. Amer. Math. Soc. 11 (1952).Google Scholar
4. Schaefer, H. H., Topological vector spaces (MacMillan New York 1964).Google Scholar