Let φ be a continuous nonzero homomorphism of the convolution algebra L1loc(R+) and also the unique extension of this homomorphism to Mloc(R+). We show that the map φis continuous in the weak* and strong opertor topologies on Mloc, considered as the dual space of Cc(R+) and as the multiplier algebra of L1loc. Analogous results are proved for homomorphism from L1 [0, a) to L1 [0, b). For each convolution algebra L1 (ω1), φ restricts to a continuous homomorphism from some L1 (ω1) to some L1 (ω2), and, for each sufficiently large L1 (ω2), φ restricts to a continuous homomorphism from some L1 (ω1) to L1 (ω2). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L1loc. We also prove results on convergent nets, continuous semigroups, and bounded sets in Mloc that we need in our study of homomorphisms.

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.

Thomas conjectured that there is no such thing as an almost prime element in a commutative radical Banach algebra. This paper shows that there is such an element. However, it still leaves open the question of whether there is an element that is almost prime, but not prime.

We study convergence in weighted convolution algebras ${{L}^{1}}\left( \omega \right)$ on ${{R}^{+}}$, with the weights chosen such that the corresponding weighted space $M\left( \omega \right)$ of measures is also a Banach algebra and is the dual space of a natural related space of continuous functions. We determine convergence factor $\eta $ for which weak$*$-convergence of $\left\{ {{\text{ }\!\!\lambda\!\!\text{ }}_{n}} \right\}$ to $\text{ }\!\!\lambda\!\!\text{ }$ in $M\left( \omega \right)$ implies norm convergence of ${{\text{ }\!\!\lambda\!\!\text{ }}_{n}}*f$ to $\text{ }\!\!\lambda\!\!\text{ *}f$ in ${{L}^{1}}\left( \omega \eta \right)$. We find necessary and sufficent conditions which depend on $\omega$ and $f$ and also find necessary and sufficent conditions for $\eta$ to be a convergence factor for all ${{L}^{1}}\left( \omega \right)$ and all $f$ in ${{L}^{1}}\left( \omega \right)$. We also give some applications to the structure of weighted convolution algebras. As a preliminary result we observe that $\eta$ is a convergence factor for $\omega$ and $f$ if and only if convolution by $f$ is a compact operator from $M\left( \omega \right)$ (or ${{L}^{1}}\left( \omega \right)$) to ${{L}^{1}}\left( \omega \eta \right)$.