Analytic crossed products and outer conjugacy classes of automorphisms of von neumann algebras. II
Analytic crossed products and outer conjugacy classes of automorphisms of von neumann algebras. II
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冯诺依曼代数自同构的解析交叉积和外共轭类。
DOI：
10.1007/bf01456189
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
K. Saito
中科院分区：
文献类型：
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作者：
P. Muhly;K. Saito
在[1]中，Arveson将一个非自伴算子代数5I（X，m，r）与概率测度空间（X，m）上的每个遍历测度保持变换τ联系起来，并证明了Ill（X，m，τ）的酉等价类决定了t的共轭类，反之亦然。也就是说，他证明了9 l（X，m，r）和2 I（X '，m'，r '）是酉等价的当且仅当τ和τ'是共轭的。代数W（X，m，τ）与由L（X）和τ构造的群测度von Neumann代数W*（X，m，τ）的一个子代数密切相关。W*（X，m，τ）的这个子代数在[7]中得到了形式化的定义，随后McAsey和作者[8-10]系统地研究了这个子代数（和其他人）在”非自伴交叉积”的名称下;现在，由于各种原因，我们称它为”解析交叉积”。需要记住的重要一点是，虽然θ i（X，m，τ）构成了τ的共轭不变量的完备集，相关的解析交叉积W*（X，m，x）也是如此，它不包含关于τ的共轭类的信息。原因很简单，对于每个这样的τ，W*（X，m，T）是一个超有限因子，并且所有这样的因子都是同构的（当然，在适当的可分性假设下）。Arveson和Josephson在[2]中改进了[1]，证明了51（X，w，t）的同构类决定了τ直到共轭。本文的目的是推广他们的分析，并将von Neumann代数的自同构的外共轭类与由该自同构构造的解析交叉积的同构类联系起来。更精确地说，设M（N）是一个vonNeumann代数，α（β）是一个 *-自同构
In [1] Arveson associated a non-self-adjoint operator algebra, 5I (X, m, r), with each ergodic measure preserving transformation, τ, on a probability measure space,(X, m), and he showed that the unitary equivalence class of ill (X, m, τ) determines the conjugacy class of t and vice versa. That is, he showed that 9l (X, m, r) and 2I (X', m', r') are unitarily equivalent if and only if τ and τ'are conjugate. The algebra'ϋ (Χ, ηι, τ) is closely related to a subalgebra of the group-measure von Neumann algebra W*(X, m, τ) constructed from L (X) and τ. This subalgebra of W*(X, m, τ) was formally defined in [7] and subsequently was studied systematically by McAsey and the authors [8-10](and by others) under the name" non-self-adjoint crossed product"; nowadays, for a variety of reasons, we call it an" analytic crossed product" The important thing to keep in mind is that while the unitary equivalence class of 9i (X, m, τ) constitutes a complete set of conjugacy invariants for τ, as does the related analytic crossed product, W*(X, m, x) contains no information about the conjugacy class of τ. The reason, quite simply, is that for each such τ, W*(X, m, T) is a hyperfinite IIt factor and all such factors are isomorphic (under the appropriate separability assumption, of course). In a subsequent paper [2], Arveson and Josephson improved on [1] by showing that the isomorphism class of 51 (X, w, t) determined τ up to conjugacy. Our objective in this paper is to extend their analysis and to relate the outer conjugacy class of an automorphism of a von Neumann algebra to the isomorphism class of the analytic crossed product constructed from the automorphism. More precisely, let M (respectively N) be a von Neumann algebra, let α (respectively β) be a*-automorphism