  Starting with a complex commutative semi-simple regular Banach algebra $A$
and an automorphism $\sigma$ of $A$, we form the crossed product of $A$ with
the integers, where the latter act on $A$ via iterations of $\sigma$. The
automorphism induces a topological dynamical system on the character space
$\Delta(A)$ of $A$ in a natural way. We prove an equivalence between the
property that every non-zero ideal in the crossed product has non-zero
intersection with the subalgebra $A$, maximal commutativity of $A$ in the
crossed product, and density of the non-periodic points of the induced system
on the character space. We also prove that every non-trivial ideal in the
crossed product always intersects the commutant of $A$ non-trivially.
Furthermore, under the assumption that $A$ is unital and such that $\Delta(A)$
consists of infinitely many points, we show equivalence between simplicity of
the crossed product and minimality of the induced system, and between primeness
of the crossed product and topological transitivity of the system.
