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Functional continuity of topological algebras with orthonormal bases. (English) Zbl 1475.46044

The paper deals with the problem of functional continuity of complex locally multiplicatively complete algebras with jointly continuous multiplication trying to partially answer some questions posed by E. A. Michael [Locally multiplicatively-convex topological algebras. Mem. Am. Math. Soc. 11. Providence, RI: American Mathematical Society (AMS) (1952; Zbl 0047.35502)].
A topological algebra is functionally continuous if every multiplicative linear functional on that algebra is continuous.
The authors prove that
(a)
every complex complete locally multiplicatively convex algebra with an orthogonal basis is functionally continuous;
(b)
every sequentially complete locally convex topological algebra with an unconditional orthonormal basis and an element fulfilling special condition (see Theorem 3.5 of the paper for details) is functionally continuous;
(c)
every unital complex \(F\)-algebra with an orthonormal basis is functionally continuous.
The authors offer also an example of a unital complete locally convex topological algebra with an unconditional orthonormal basis which is not functionally continuous.
Reviewer: Mart Abel (Tartu)

MSC:

46H40 Automatic continuity
46H05 General theory of topological algebras
46H20 Structure, classification of topological algebras

Citations:

Zbl 0047.35502
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Full Text: DOI

References:

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