Invariant states and ergodic dynamical systems on \(W^*\)-algebras. (English) Zbl 0765.46052
Let \(M\) be a \(\sigma\)-finite \(W^*\)-algebra, \(G\) an amenable semigroup and \(\alpha\) a representation of \(G\) into the set of linear normal positive identity preserving mappings on \(M\). A normal faithful state \(\omega\) of \(M\) is said to be \(G\) invariant if \(\omega\circ\alpha_ g=\omega\) for all \(g\) in \(G\).
Theorem: The following conditions are equivalent:
(i) there exists a normal faithful \(G\) invariant state of \(M\);
(ii) \(\overline{\{\alpha_ g:\;g\in G\}}\) consists of faithful positive linear mappings of \(M\); and
(iii) \(0\not\in\overline{\{\alpha_ g e:\;g\in G\}}\) for each nonzero projection \(e\) in \(M\).
Theorem: The following conditions are equivalent:
(i) there exists a normal faithful \(G\) invariant state of \(M\);
(ii) \(\overline{\{\alpha_ g:\;g\in G\}}\) consists of faithful positive linear mappings of \(M\); and
(iii) \(0\not\in\overline{\{\alpha_ g e:\;g\in G\}}\) for each nonzero projection \(e\) in \(M\).
Reviewer: S.Sankaran (London)
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