Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T15:40:28.133Z Has data issue: false hasContentIssue false

Nonstandard Ideals from Nonstandard Dual Pairs for L1(ω) and l1(ω)

Published online by Cambridge University Press:  20 November 2018

C. J. Read*
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Banach convolution algebras ${{l}^{1}}(\omega )$ and their continuous counterparts ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$ are much studied, because (when the submultiplicative weight function $\omega $ is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of “nice” weights $\omega $, the only closed ideals they have are the obvious, or “standard”, ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in ${{l}^{1}}(\omega )$. His proof was successfully exported to the continuous case ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$ by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in ${{l}^{1}}(\omega )$ and ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$. The new proof is based on the idea of a “nonstandard dual pair” which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$ containing functions whose supports extend all the way down to zero in ${{\mathbb{R}}^{+}}$, thereby solving what has become a notorious problem in the area.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Dales, H. G., Banach algebras and automatic continuity, London Mathematical Society Monographs New Series 24, Clarendon Press, New York, 2000.Google Scholar
[2] Dales, H. G. and McClure, J. P., Nonstandard ideals in radical convolution algebras on a half-line. Canad. J. Math. 39(1987), no. 2, 309321.Google Scholar
[3] Domar, Y., Cyclic elements under translation in weighted L1 spaces on ℝ+ , Ark. Mat 19(1981), no. 1, 137144.Google Scholar
[4] Ghahramani, F. and Grabiner, S., Convergence factors and compactness in weighted convolution algebras. Canad J. Math. 54(2002), no. 2, 303323.Google Scholar
[5] Grabiner, S. and Thomas, M. P., Nonunicellular strictly cyclic quasinilpotent shifts on Banach spaces. J. Operator Theory 13(1985), 163170.Google Scholar
[6] Thomas, M. P., A nonstandard ideal of a radical Banach algebra of power series. Acta Math. 152(1984), no. 3-4, 199217.Google Scholar