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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Topological dynamics and $ C\sp{\ast} $-algebras

Author: William L. Green
Journal: Trans. Amer. Math. Soc. 210 (1975), 107-121
MSC: Primary 46L05; Secondary 54H15
MathSciNet review: 0383091
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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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Keywords: $ {C^ \ast }$-algebra, transformation group, automorphism, uniformly almost periodic, uniquely ergodic, (finite) trace, central $ {C^ \ast }$-algebra
Article copyright: © Copyright 1975 American Mathematical Society