A-Valued Semicircular Systems

Abstract

To a von Neurnann algebra A and a set of linear maps ηij:A→A, i, j∈I such that a↦(ηij)ij∈I: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, 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.