  If $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$,
then a Banach algebra $\ell^1(\Sigma)$ of crossed product type is naturally
associated with this topological dynamical system $\Sigma=(X,\sigma)$. If $X$
consists of one point, then $\ell^1(\Sigma)$ is the group algebra of the
integers.
  We study the algebraically irreducible representations of $\ell^1(\Sigma)$ on
complex vector spaces, its primitive ideals and its structure space. The finite
dimensional algebraically irreducible representations are determined up to
algebraic equivalence, and a sufficiently rich family of infinite dimensional
algebraically irreducible representations is constructed to be able to conclude
that $\ell^1(\Sigma)$ is semisimple. All primitive ideals of $\ell^1(\Sigma)$
are selfadjoint, and $\ell^1(\Sigma)$ is Hermitian if there are only periodic
points in $X$. If $X$ is metrisable or all points are periodic, then all
primitive ideals arise as in our construction. A part of the structure space of
$\ell^1(\Sigma)$ is conditionally shown to be homeomorphic to the product of a
space of finite orbits and $\mathbb T$. If $X$ is a finite set, then the
structure space is the topological disjoint union of a number of tori, one for
each orbit in $X$. If all points of $X$ have the same finite period, then it is
the product of the orbit space $X/\mathbb Z$ and $\mathbb T$. For rational
rotations of $\mathbb T$, this implies that the structure space is homeomorphic
to $\mathbb T^2$.
