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The spectral radius in a locally convex algebra
Published online by Cambridge University Press: 24 October 2008
Extract
In (1), Allan introduced the concept of the spectrum of an element of a locally convex algebra, and developed a spectral theory for pseudo-complete algebras. In a commutative Banach algebra the spectral radius is a continuous seminorm, and so it is natural to investigate continuity properties of the spectral radius in various classes of locally convex algebra. Continuity is too strong a condition to be expected in any general case, and an interesting property to investigate appears to be lower semi-continuity. We shall show easily that in a commutative pseudo-complete locally m-convex algebra (in the sense of (4)) the spectral radius is lower semi-continuous. We shall then exhibit a commutative complete metrizable algebra in which lower semi-continuity fails to hold.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 3 , November 1974 , pp. 493 - 496
- Copyright
- Copyright © Cambridge Philosophical Society 1974
References
REFERENCES
(1)Allan, G. R.A spectral theory for locally convex algebras. Proc. London Math. Soc. 15 (1965), 399–421.CrossRefGoogle Scholar
(2)Dixon, P. G.An embedding theorem for commutative B0-algebras. Studia Math. 41 (1972), 163–168.CrossRefGoogle Scholar
(3)Grothendieck, A.Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16 (1955).Google Scholar
(4)Michael, E. A.Locally multiplicatively-convex topological algebras. Mem. Amer. Math. Soc. 11 (1952).Google Scholar