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Radical Q-algebras

Published online by Cambridge University Press:  18 May 2009

P. G. Dixon
P. G. Dixon
Affiliation:
Department of Pure Mathematics, Hicks Building, The University, Sheffield, S3 7RH
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The purpose of this paper is to exhibit various Q-algebras (quotients of uniform algebras) which are Jacobson radical. We begin by noting easy examples of nilpotent Q-algebras and Q-algebras with dense nil radical. Then we describe two ways of constructing semiprime, Jacobson radical Q-algebras. The first is by directly constructing a uniform algebra and an ideal. This produces a nasty Q-algebra as the quotient of a nice uniform algebra (in the sense that it is a maximal ideal of R(X) for some X ⊆ ℂ). The second way is by using results of Craw and Varopoulos to show that certain weighted sequence algebras are Q-algebras. In fact we show that a weighted sequence algebra is Q if the weights satisfy (i) w(n+1)/w(n)↓0 and (ii) (w(n+l)/w(n))∊lp for some p≧ 1, but may be non-Q if either (i) or (ii) fails. This second method produces nice Q-algebras which are quotients of rather horrid uniform algebras as constructed by Craw's Lemma.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 17 , Issue 2 , July 1976 , pp. 119 - 126
Copyright
Copyright © Glasgow Mathematical Journal Trust 1976

References

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