  The Banach space $\ell_1(\Z)$ admits many non-isomorphic preduals, for
example, $C(K)$ for any compact countable space $K$, along with many more
exotic Banach spaces. In this paper, we impose an extra condition: the predual
must make the bilateral shift on $\ell_1(\Z)$ weak$^*$-continuous. This is
equivalent to making the natural convolution multiplication on $\ell_1(\Z)$
separately weak*-continuous and so turning $\ell_1(\Z)$ into a dual Banach
algebra. We call such preduals \emph{shift-invariant}. It is known that the
only shift-invariant predual arising from the standard duality between $C_0(K)$
(for countable locally compact $K$) and $\ell_1(\Z)$ is $c_0(\Z)$. We provide
an explicit construction of an uncountable family of distinct preduals which do
make the bilateral shift weak$^*$-continuous. Using Szlenk index arguments, we
show that merely as Banach spaces, these are all isomorphic to $c_0$. We then
build some theory to study such preduals, showing that they arise from certain
semigroup compactifications of $\Z$. This allows us to produce a large number
of other examples, including non-isometric preduals, and preduals which are not
Banach space isomorphic to $c_0$.
