Analytic crossed products and outer conjugacy classes of automorphisms of von Neumann algebras. II (Q1819328)

Let M be a von Neumann algebra and let \(\alpha\) be a *-automorphism of M. The analytic crossed product of M with \(\alpha\), \(M\times _{\alpha}{\mathbb{Z}}_ +\), is constructed from M and the non-negative powers of \(\alpha\) in much the same way as the von Neumann algebra crossed product of M with \(\alpha\) is constructed from M and the entire group \(\{\alpha ^ n\}_{n\in {\mathbb{Z}}}\). \(M\times _{\alpha}{\mathbb{Z}}_ +\) should be viewed as a noncommutative version of \(H^{\infty}({\mathbb{T}}).\) In this note we show that under suitable hypotheses, if two analytic crossed products are isomorphic then the associated automorphisms are outer conjugate. Our results generalize results by Arveson and by Arveson and Josephson who considered the case when the coefficient algebra M is abelian. This note also generalizes our earlier study on the same subject. The hypotheses in that work are weakened considerably while new and simpler proofs are given.