## Weighted convolution algebras on subsemigroups of the real line

### Volume 459 / 2009

#### Abstract

\def\SAI{strongly Arens irregular}\def\q{{\sym Q}^{+\bullet}}In this memoir, we shall consider weighted convolution algebras on discrete groups and semigroups, concentrating on the group $(\Q, +)$ of rational numbers, the semigroup $(\q, +)$ of strictly positive rational numbers, and analogous semigroups in the real line~$\R$. In particular, we shall discuss when these algebras are Arens regular, when they are {\SAI}, and when they are neither, giving a variety of examples. We introduce the notion of `weakly diagonally bounded' weights, weakening the known concept of `diagonally bounded' weights, and thus obtaining more examples. We shall also construct an example of a weighted convolution algebra on~$\sym N$ that is neither Arens regular nor~{\SAI}, and an example of a weight $\omega$ on~$\q$ such that $\liminf_{s\to 0+}\omega(s) =0$.