Abstract
To a von Neurnann algebra A and a set of linear maps η ij : A → A , i , j ∈ I such that a ↦( η ij ) ij ∈ I : A → A ⊗ B ( l 2 ( 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 X i , i ∈ I , 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.
Published Version
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