TRANSACTIONS OF THE AMERICAN MATHEMATICALSOCIETY Volume 210, 1975 TOPOLOGICAL DYNAMICS ANDC*-ALGEBRAS(1 ) BY WILLIAM L. GREEN ABSTRACT. If G is a group of automorphisms of a C -algebra A with iden- tity, then G acts in a natural way as a transformation group on the state space S(A) of A. Moreover, this action is uniformly almost periodic if and only if G has compact pointwise closure in the space of all maps of A into A. Considera- tion of the enveloping semigroup of (S(A), G) shows that, in this case, this point- wise closure G is a compact topological group consisting of automorphisms of A. The Haar measure on G is used to define an analogue of the canonical center- valued trace on a finite von Neumann algebra. If A possesses a sufficiently large group Go of inner automorphisms such that (S(A), Gq) is uniformly almost peri- odic, then A is a central C -algebra. The notion of a uniquely ergodic system is applied to give necessary and sufficient conditions that an approximately finite dimensional C -algebra possess exactly one finite trace. Introduction. The purpose of this paper is to apply some ideas from topo- logical dynamics to the study of C*-algebras. If X is a compact Hausdorff space and (X, V) is a topological transformation group, then T has a natural representa- tion as a group of automorphisms of the commutative C*-algebra C(X): for t G T and/GCLY) put itf)(x) = f(xt), xGX. It is often possible to express properties of (X, T) in terms of the system (r, C(X)); for example, (X, T) is uniformly almost periodic iff for each /G C(X), the set [tf: t G T} is relatively compact in C(X). If A is an arbitrary C*-algebra with identity and G is a group of automorphisms of A, we may view the pair (G, A) as a noncommutative version of (r, C(X)). We shall see that some of the relationships between (X, V) and (r, C(X)) have noncommutative analogues, and that these analogues can be used to obtain information about the structure of cer- tain C*-algebras. Presented to the Society, November 8, 1974; received by the editors July 1, 1974. AMS(MOS)subject classifications (1970). Primary 46L05; Secondary S4H15, 22D2S, 22D45, 46L25. Key words and phrases. C -algebra, transformation group, automorphism, uniformly almost periodic, uniquely ergodic, (finite) trace, central C -algebra. (!)This work was the author's doctoral dissertation at the University of Pennsylvania, and he would like to thank Edward G. Effros, under whose supervision the research was car- ried out, for his encouragement and his patient and very helpful advice. He is also indebted to Erling Stürmer for a number of helpful conversations concerning the material discussed in §§2 and 3. Copyright © 1975, American Mathematical Society 107 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use