  We discuss special properties of the spaces of characters and positive
definite functions, as well as their associated dynamics, for arithmetic groups
of product type. Axiomatizing these properties, we define the notions of
charmenability and charfiniteness and study their applications to the
topological dynamics, ergodic theory and unitary representation theory of the
given groups. To do that, we study singularity properties of equivariant normal
ucp maps between certain von Neumann algebras. We apply our discussion also to
groups acting on product of trees.
