Topological dynamics and 𝐶*-algebras

Green, William L.

doi:10.1090/S0002-9947-1975-0383091-6

Topological dynamics and $C^{\ast }$-algebras
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by William L. Green PDF

Trans. Amer. Math. Soc. 210 (1975), 107-121 Request permission

Abstract:

If $G$ is a group of automorphisms of a ${C^ \ast }$-algebra $A$ with identity, 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 $A$. Consideration of the enveloping semigroup of $(S(A),G)$ shows that, in this case, this pointwise closure $\bar G$ is a compact topological group consisting of automorphisms of $A$. The Haar measure on $\bar G$ 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 ${G_0}$ of inner automorphisms such that $(S(A),{G_0})$ is uniformly almost periodic, then $A$ is a central ${C^ \ast }$-algebra. The notion of a uniquely ergodic system is applied to give necessary and sufficient conditions that an approximately finite dimensional ${C^ \ast }$-algebra possess exactly one finite trace.

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