Topological dynamics on $C^ *$-algebras

Abstract:

Dynamical properties of a group of homeomorphisms of a compact Hausdorff space $X$ can be interpreted in terms of the commutative ${C^\ast }$-algebra $C(X)$. We investigate a noncommutative topological dynamics extending dynamical concepts to the context of a group of automorphisms on a general ${C^\ast }$-algebra with unit. Such concepts as minimality, almost periodicity, and point-wise almost periodicity are extended to this situation. Theorems are obtained extending commutative dynamical results and relating the noncommutative dynamics to the transformation groups induced on the state space and the weak* closure of the pure states. We show, for example, that the group acts almost periodically on the ${C^\ast }$-algebra if and only if each of these induced transformation groups is almost periodic.