Topological Dynamics and C ∗ -Algebras
William L. Green · Transactions of the American Mathematical Society · 1975
If G is a group of au.tomorphisms of a C -algebra A with identityt then G acta in a natural way as a transformation group on the state space S(A) of A • Moreover, this action is uniformly almoat periodic iff G has compact pointwise closure in the space of all maps of A into A~ Consideration of the enveloping semigroup of (S{A),G) shows that in this case, this poin~ wise closure ~ is a compact topological group consisting of automorphiams of A. The Haar measure on ~ is used to define an analogue of the canonical center~valued tr~ce in a finite von Neumann algebra.If A possesses a sufficiently large group G of inner autoworphiems such that 0 (S(.A) 7 G 0 ) is uni:f'ornlly al-•l!moat periodici then A is a central C -algebra.The notion of a uniquely ergodic system ia applied to give necessary and sufficient conditions that an approximately finite exactly one finite trace.* C -algebra possess ' periodic iff for ea.ch f E C{X) ~• the Sf'lt ( tf: t E r} is relatively compact C(X) If A * in :ts Qn arbitrary c -algebra with identity and G is a group of automorphisms of A , we may view the pair (G.A) as a. :nort-•comnrutative Yersion of (ftC(X)) • We shall see that some of the relationships between (X 1 r) •and (f 1 C(X)) have non-commutative analogues, and that these an:if.Joguea can be used to * obtain information about the structu:re of certain C ~•algebras.