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A strong containment property for discrete amenable groups of automorphisms on $ W\sp \ast$ algebras


Author: Edmond E. Granirer
Journal: Trans. Amer. Math. Soc. 297 (1986), 753-761
MSC: Primary 46L30; Secondary 43A07, 46L40, 46L55
DOI: https://doi.org/10.1090/S0002-9947-1986-0854097-2
MathSciNet review: 854097
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Abstract: Let $ G$ be a countable group of automorphisms on a $ {W^{\ast}}$ algebra $ \mathcal{M}$ and let $ {\phi _0}$ be a $ {w^{\ast}}{G_\delta }$ point of the set of $ G$ invariant states on $ \mathcal{M}$ which belong to $ {w^{\ast}}\operatorname{cl} \operatorname{Co} E$, where $ E$ is a set of (possibly pure) states on $ \mathcal{M}$. If $ G$ is amenable, then the cyclic representation $ {\pi _{{\phi _0}}}$ corresponding to $ {\phi _0}$ is contained in $ \{ \oplus {\pi _\phi };\phi \in E\} $. This property characterizes amenable groups. Related results are obtained.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0854097-2
Article copyright: © Copyright 1986 American Mathematical Society

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