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.