Asymptotic Galois correspondence for discrete amenable group actions on factors (Q1185748)

The present article is a contribution to the theory of group actions on factors. A result of the first named author [J. Oper. Theory 21, 297-314 (1989; Zbl 0715.46034)] is extended to non-type I approximately finite dimensional factors. The following theorem is proved: Let \({\mathcal R}\) be an AFD factor of non-type I and \(\alpha\) be an ultrafree action of a discrete amenable group \(G\). For an automorphism \(\Theta\) of \({\mathcal R}\), the following two conditions are equivalent: (i) \(\Theta\) is of the form \(\Theta=\alpha_{g_ 0}\) for \(g_ 0\in G\); (ii) If \(\{x_ n\}\) is a bounded sequence in \({\mathcal R}\) such that \(\{\alpha_ g(x_ n)-x_ n\}\) converges to zero \(\sigma^*\)-strongly for every \(g\in G\), then \(\{\Theta(x_ n)-x_ n\}\) also converges to zero \(\sigma^*\)-strongly.