Automorphisms of Araki-Woods factors of type \(III_ 1\) (Q794240)

An AF-factor is called Araki-Woods factor if it is decomposed into an infinite tensor product of finite \(I_ n\)-type factors. It is known that all Araki-Woods factors of type \(III_ 1\) are isomorphic. The main results of the article: Let Int \(R_{\infty}\) be a group of inner automorphisms of the Araki- Woods \(III_ 1\)-type factor \(R_{\infty}\), and let \(\overline{Int R}_{\infty}\) be its closure in \(Aut R_{\infty},\) then \(Aut R_{\infty}=\overline{Int R}_{\infty}.\) Let \(\Gamma\) be an amenabel countable group of exterior automorphisms of \(R_{\infty}\) such that the crossed product \(W^*(R_{\infty},\Gamma)\) is a \(III_ 1\)-factor, then \(W^*(R_{\infty},\Gamma)\cong R_{\infty}.\)