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Nonstandard Ideals from Nonstandard Dual Pairs for $L^1(\omega)$ and $l^1(\omega)$

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 Printed: Aug 2006
  • C. J. Read
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The Banach convolution algebras $l^1(\omega)$ and their continuous counterparts $L^1(\bR^+,\omega)$ 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 $L^1(\bR^+,\omega)$ 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 $L^1(\bR^+,\omega)$. 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 $L^1(\bR^+,\omega)$ containing functions whose supports extend all the way down to zero in $\bR^+$, thereby solving what has become a notorious problem in the area.
Keywords: Banach algebra, radical, ideal, standard ideal, semigroup Banach algebra, radical, ideal, standard ideal, semigroup
MSC Classifications: 46J45, 46J20, 47A15 show english descriptions Radical Banach algebras
Ideals, maximal ideals, boundaries
Invariant subspaces [See also 47A46]
46J45 - Radical Banach algebras
46J20 - Ideals, maximal ideals, boundaries
47A15 - Invariant subspaces [See also 47A46]

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