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On the Invariance of the Spectrum in Locally m-Convex Algebras

Published online by Cambridge University Press:  20 November 2018

R. M. Brooks
R. M. Brooks*
Affiliation:
University of Minnesota, Minneapolis, Minnesota
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In this paper we consider two closely related problems concerning a complete locally m-convex (LMC) algebra A with identity. Let a be a fixed element of A, and let P(a) be the smallest closed subalgebra containing a and 1. If B is any subalgebra containing a and 1, we let σ(a; B) denote the spectrum of a as an element of B. (I) Describe the set σ(a; P(a)) in terms of σ(a; A). (II) Give necessary and sufficient conditions in order that σ (a; B) = σ(a; A) for every closed subalgebra B of A which contains a and 1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 658 - 664
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

The research for this paper was supported in part by NSF Grant GP5707.

References

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3. Rickart, C. E., General theory of Banach algebras (Van Nostrand, New York, 1960).Google Scholar