A-Valued Semicircular Systems,
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- 作者：Shlyakhtenko ; Dimitri
- 刊名：Journal of Functional Analysis
- 出版年：1999
- 出版时间：August 1, 1999
- 年：1999
- 卷：166
- 期：1
- 页码：1-47
- 全文大小：341 K
文摘
To a von Neurnann algebra A and a set of linear maps ηij:A→A, i, j
I such that a
(ηij)ijI:A→A
B(l2(I)) is normal and completely positive, we associate a von Neumann algebra Φ(A, η). This von Neumann algebra is generated by A and an A-valued semicircular system Xi, iI, associated to η. In many cases there is a faithful conditional expectation E:Φ(A, η)→A; if A is tracial, then under certain assumptions on η, Φ(A, η) also has a trace. One can think of the construction Φ(A, η) as an analogue of a crossed product construction. We show that most known algebras arising in free probability theory can be obtained from the complex field by iterating the construction Φ. Of a particular interest are free Krieger algebras, which, by analogy with crossed products and ordinary Krieger factors, are defined to be algebras of the form Φ(L∞[0, 1], η). The cores of free Araki–Woods factors are free Krieger algebras. We study the free Krieger algebras and as a result obtain several non-isomorphism results for free Araki–Woods factors. As another source of classification results for free Araki–Woods factors, we compute the τ invariant of Connes for free products of von Neumann algebras. This computation generalizes earlier work on computation of T, S, and Sd invariants for free product algebras.
I such that a(ηij)ijI:A→AB(l2(I)) is normal and completely positive, we associate a von Neumann algebra Φ(A, η). This von Neumann algebra is generated by A and an A-valued semicircular system Xi, iI, associated to η. In many cases there is a faithful conditional expectation E:Φ(A, η)→A; if A is tracial, then under certain assumptions on η, Φ(A, η) also has a trace. One can think of the construction Φ(A, η) as an analogue of a crossed product construction. We show that most known algebras arising in free probability theory can be obtained from the complex field by iterating the construction Φ. Of a particular interest are free Krieger algebras, which, by analogy with crossed products and ordinary Krieger factors, are defined to be algebras of the form Φ(L∞[0, 1], η). The cores of free Araki–Woods factors are free Krieger algebras. We study the free Krieger algebras and as a result obtain several non-isomorphism results for free Araki–Woods factors. As another source of classification results for free Araki–Woods factors, we compute the τ invariant of Connes for free products of von Neumann algebras. This computation generalizes earlier work on computation of T, S, and Sd invariants for free product algebras.