We develop methods for studying transition operators on metric spaces that
are invariant under a co-compact group which acts properly. A basic
requirement is a decomposition of such operators with respect to the group
orbits. We then introduce "reduced" transition operators on the compact
factor space whose norms and spectral radii are upper bounds for the
$L^p$-norms and spectral radii of the original operator. If the group is
amenable then the spectral radii of the original and reduced operators
coincide, and under additional hypotheses, this is also sufficient for
amenability. Further bounds involve the modular function of the group.
In this framework, we prove among other things that the bottom of the
spectrum of the Laplacian on a co-compact Riemannian manifold is 0 if
and only if the group is amenable and unimodular. The same result holds
for Euclidean simplicial complexes. On a geodesic, proper metric space
with co-compact isometry group action, the averaging operator over balls
with a fixed radius has norm equal to 1 if and only if the group is
amenable and unimodular. The technique also allows explicit computation of
spectral radii when the group is amenable.