×

A remark on functional continuity of certain Fréchet algebras. (English) Zbl 0717.46046

Let A be a Fréchet algebra. Denote by S(A) the set of all non zero multiplicative linear functionals and by M(A) the set of all continuous members of S(A). The author shows that if there are \(f_ 1,f_ 2,...,f_ n\in A\) such that for each \(x\in M(A)\) the set \(\{\) \(y\in M(A):\) \(y(f_ i)=x(f_ i)\), \(i=1,...,n\}\) is at most countable, then \(S(A)=M(A)\). In particular if the mapping \(x\to (x(f_ 1),...,x(f_ n))\) is discrete then \(S(A)=M(A)\) (theorem 1 and corollary 1). He also describes a sort of reduction principle for the functional continuity problem. For instance if \(M(A)=\cup_{i\in I}L_ i\), with each \(L_ i\) open and compact, then A is functionally continuous (lemma 2).
Reviewer: J.Ludwig (Metz)

MSC:

46J25 Representations of commutative topological algebras
46H40 Automatic continuity
PDFBibTeX XMLCite
Full Text: DOI EuDML