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The spectral radius in a locally convex algebra

Published online by Cambridge University Press:  24 October 2008

A. W. Wood
A. W. Wood
Affiliation:
Trinity College, Cambridge
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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

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References

REFERENCES

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