If X is a compact Hausdorff space, supplied with a homeomorphism, then a
crossed product involutive Banach algebra is naturally associated with these
data. If X consists of one point, then this algebra is the group algebra of the
integers. In this paper, we study spectral synthesis for the closed ideals of
this associated algebra in two versions, one modeled after C(X), and one
modeled after the group algebra of the integers. We identify the closed ideals
which are equal to (what is the analogue of) the kernel of their hull, and
determine when this holds for all closed ideals, i.e., when spectral synthesis
holds. In both models, this is the case precisely when the homeomorphism has no
periodic points.Comment: 28 page
