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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

REFERENCES

1.Bonsall, F. F. and Duncan, J., Complete normed algebras (Springer-Verlag, 1973).CrossRefGoogle Scholar
2.Craw, I. G. and Young, N. J., Regularity of multiplication in weighted group and semigroup algebras, Quart. J. Math. Oxford (3) 25 (1974), 351–358.CrossRefGoogle Scholar
3.Davie, A. M., Quotient algebras of uniform algebras, J. Lonon Math. Soc. (2) 7 (1973), 31–40.CrossRefGoogle Scholar
4.Dixon, P. G., Semiprime Banach algebras, J. Lonon Math. Soc. 6 (1973), 676–678.CrossRefGoogle Scholar
5.McKissick, R., A nontrivial normal sup norm algebra, Bull. Amer. Math. Soc. 69 (1963), 391–395.CrossRefGoogle Scholar
6.Stout, E. L., The theory of uniform algebras (Bogden and Quigley, 1971).Google Scholar
7.Varopoulos, N. Th., Some remarks on Q-algebras, Ann. Inst. Fourier (Grenoble) 22, 4 (1972), 1–11.CrossRefGoogle Scholar
8.Zygmund, A., Trigonometric series vol. I (C.U.P., 1959).Google Scholar