A strong containment property for discrete amenable groups of automorphisms on 𝑊* algebras

Granirer, Edmond E.

doi:10.1090/S0002-9947-1986-0854097-2

A strong containment property for discrete amenable groups of automorphisms on $W^ \ast$ algebras
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Trans. Amer. Math. Soc. 297 (1986), 753-761 Request permission

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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