  We survey recent results regarding the study of dynamical properties of the
space of positive definite functions and characters of higher rank lattices.
These results have several applications to ergodic theory, topological
dynamics, unitary representation theory and operator algebras. The key novelty
in our work is a dynamical dichotomy theorem for equivariant faithful normal
unital completely positive maps between noncommutative von Neumann algebras and
the space of bounded measurable functions defined on the Poisson boundary of
semisimple Lie groups.
