Nonstandard Ideals from Nonstandard Dual Pairs for L1(ω) and l 1(ω)

Read, C. J.

Canadian journal of mathematics, Tome 58 (2006) no. 4, pp. 859-876

Résumé

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.

Détail

DOI : 10.4153/CJM-2006-035-6

Mots-clés : 46J45, 46J20, 47A15, Banach algebra, radical, ideal, standard ideal, semigroup

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