  We introduce a notion of noncommutative Choquet simplex, or briefly an nc
simplex, that generalizes the classical notion of a simplex. While every
simplex is an nc simplex, there are many more nc simplices. They arise
naturally from C*-algebras and in noncommutative dynamics. We characterize nc
simplices in terms of their geometry and in terms of structural properties of
their corresponding operator systems.
  There is a natural definition of nc Bauer simplex that generalizes the
classical definition of a Bauer simplex. We show that a compact nc convex set
is an nc Bauer simplex if and only if it is affinely homeomorphic to the nc
state space of a unital C*-algebra, generalizing a classical result of Bauer
for unital commutative C*-algebras.
  We obtain several applications to noncommutative dynamics. We show that the
set of nc states of a C*-algebra that are invariant with respect to the action
of a discrete group is an nc simplex. From this, we obtain a noncommutative
ergodic decomposition theorem with uniqueness.
  Finally, we establish a new characterization of discrete groups with
Kazhdan's property (T) that extends a result of Glasner and Weiss.
Specifically, we show that a discrete group has property (T) if and only if for
every action of the group on a unital C*-algebra, the set of invariant states
is affinely homeomorphic to the state space of a unital C*-algebra.
