Second Duals of Beurling Algebras

Let $A$ be a Banach algebra, with second dual space $A''$. We propose to study the space $A''$ as a Banach algebra. There are two Banach algebra products on $A''$, denoted by $\,\Box\,$ and $\,\Diamond\,$. The Banach algebra $A$ is Arens regular if the two products $\Box$ and $\Diamond$ coincide on $A''$. In fact, $A''$ has two topological centres denoted by $\mathfrak{Z}^{(1)}_t(A'')$ and $\mathfrak{Z}^{(2)}_t(A'')$ with $A \subset \mathfrak{Z}^{(j)}_t(A'')\subset A''\;\,(j=1,2)$, and $A$ is Arens regular if and only if $\mathfrak{Z}^{(1)}_t(A'')=\mathfrak{Z}^{(2)}_t(A'')=A''$. At the other extreme, $A$ is strongly Arens irregular if $\mathfrak{Z}^{(1)}_t(A'')=\mathfrak{Z}^{(2)}_t(A'')=A$. We shall give many examples to show that these two topological centres can be different, and can lie strictly between $A$ and $A''$. We shall discuss the algebraic structure of the Banach algebra $(A'',\,\Box\,)$; in particular, we shall seek to determine its radical and when this algebra has a strong Wedderburn decomposition. We are also particularly concerned to discuss the algebraic relationship between the two algebras $(A'',\,\Box\,)$ and $(A'',\,\Diamond\,)$. Most of our theory and examples will be based on a study of the weighted Beurling algebras $L^1(G,\omega)$, where $\omega$ is a weight function on the locally compact group $G$. The case where $G$ is discrete and the algebra is ${\ell}^{\,1}(G, \omega )$ is particularly important. We shall also discuss a large variety of other examples. These include a weight $\omega$ on $\mathbb{Z}$ such that $\ell^{\,1}(\mathbb{Z},\omega)$ is neither Arens regular nor strongly Arens irregular, and such that the radical of $(\ell^{\,1}(\mathbb{Z},\omega)'', \,\Box\,)$ is a nilpotent ideal of index exactly $3$, and a weight $\omega$ on $\mathbb{F}_2$ such that two topological centres of the second dual of $\ell^{\,1}(\mathbb{F}_2, \omega)$ may be different, and that the radicals of the two second duals may have different indices of nilpotence.