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Closed ideals in the Banach algebra ℓ1(ω) when ω is ∈-star shaped

Published online by Cambridge University Press:  20 January 2009

Marc P. Thomas
Marc P. Thomas
Affiliation:
Mathematics Department, California State University, Bakersfield, 9001 Stockdale Highway, Bakersfield, California 93311–1099, USA
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Provided that the weight function ω satisfies certain submultiplicative and decay conditions, the discrete convolution algebra ℓ1(ω) becomes a commutative radical Banach algebra with identity adjoined. There are obvious closed ideals in ℓ1(ω) and these are denoted standard ideals. Earlier results of Thomas, strengthened by Yakubovich and Domar, showed that if the weight ω is star-shaped then all closed ideals are standard. Consequently, the closed ideal generated by any element f in ℓ1(ω) must be standard.

The requirement that ω be star-shaped (essentially that ω(n)1/n must decrease to zero) is somewhat restrictive in that no local maxima of ω(n)1/n are allowed. We generalize this previous result to apply to the larger class of ε-star shaped weights (0 < ε ≤ 1) which allow such local maxima. If f is a non-zero element on ℓ1(ω) we let the integer α(f) = k0 denote the index of its first non-zero term. We introduce the concept of an ε-peak point for k0. If ε = 1 then ω is star-shaped in the usual sense and there are an infinite number of 1-peak points for any k0. Although this latter fact may fail if 0 < ε < 1, if ω(n)1/n tends to zero sufficiently quickly (dependent on k0 and ε) there will always be an infinite number of ε-peak points for k0.

Our main result is that if ω is an ε-star shaped weight, if f is an non-zero element of ℓ1(ω), if α(f) = k0, and if the number of ε-peak points for k0 is infinite, then the closed ideal generated by f is standard.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , Issue 1 , February 1998 , pp. 161 - 176
Copyright
Copyright © Edinburgh Mathematical Society 1998

References

REFERENCES

1.Bade, W. G. and Dales, H. G., Norms and ideals in radical convolution algebras, J. Funct. Anal., to appear.Google Scholar
2.Domar, Y., On the unicellularity of ℓp(ω), Monatsh. Math. 103 (1987), 103–113.CrossRefGoogle Scholar
3.Grabiner, S., Weighted shifts and Banach algebras of power series, Amer. J. Math. 97 (1975), 16–42.CrossRefGoogle Scholar
4.Laursen, K. B., Ideal structure in radical sequence algebras, in Radical Banach algebras and automatic continuity (Proceedings Long Beach, 1981, Springer-Verlag, 975).Google Scholar
5.Nikolskii, N. K., Selected problems of weighted approximation and spectral analysis, Proc. Steklov Inst. Math. 120 (1974) (AMS translation, 1976).Google Scholar
6.Shields, A. L., Weighted shift operators and analytic function theory, in Topics in Operator Theory (ed. Pearcy, C., AMS Surveys 13, 1974).CrossRefGoogle Scholar
7.Thomas, M. P., Approximation in the radical algebra ℓ1(ω) when {ω} is star-shaped, in Radical Banach algebras and automatic continuity (Proceedings Long Beach, 1981, Springer-Verlag, 975).Google Scholar
8.Thomas, M. P., A non-standard ideal of a radical Banach algebra of power series, Acta Math. 152 (1984), 199–217.CrossRefGoogle Scholar
9.Thomas, M. P., Quasinilpotent strictly cyclic unilateral weighted shift operators on ℓp which are not unicellular, Proc. London Math. Soc. (3) 51 (1985), 127–145.CrossRefGoogle Scholar
10.Yakubovich, D. V., Invariant subspaces of weighted shift operators, Zap. Nauk Sem. LOMI 141 (1985), 100–143.Google Scholar