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.