Duality type results and ergodic actions of simple Lie groups on operator algebras (Q1173649)

Let \(G\) be a locally compact group, and let \(H\) be a closed subgroup of \(G\). Let \(\beta\) be an action of \(H\) on a \(C^*\)-algebra. Denote by \(\alpha\) the induced action of \(G\) on the induced \(C^*\)-algebra. It is proved that \(\beta\) is topologically transitive if and only \(\alpha\) is. This can be viewed as a noncommutative extension of the Mackey-Moore duality principle. Suppose now \(G\) is a simple Lie group and \(H\) is noncompact. The following extension of Moore's ergodicity theorem to \(W^*\)-dynamical systems is proved. Let \(\alpha\) be an ergodic action on \(G\) on a \(W^*\)- algebra \(M\) with a \(G\)-invariant state. Then the restriction \(\alpha| H\) is ergodic.