A-valued semicircular systems

To a von Neurnann algebra A and a set of linear maps eta(ij): A --> A; i, j is an element of I such that a --> (eta(ij))(ij is an element of I): A --> A x B(t(2)(I)) is normal and completely positive, we associate a von Neuma nn algebra Phi(A, eta) This von Neumann algebra is generated by A and an A- valued semicircular system X-i, i is an element of I, associated to eta. In many cases there is a faithful conditional expectation E: Phi(A, eta) --> A; if A is tracial, then under certain assumptions on eta, Phi(A, eta) also has a trace. One can think of the construction Phi(A, eta) 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 itera ting the construction Phi. Of a particular interest are free Krieger algebr as, which, by analogy with crossed products and ordinary Krieger factors, a re defined to be algebras of the form Phi(L-infinity[0, 1], eta) The cores of free Araki-Woods factors are free Krieger algebras. We study the free Kr ieger algebras and as a result obtain several non-isomorphism results for F ree Araki-Woods factors. As another source of classification results for fr ee Araki-Woods factors, we compute the tau invariant of Connes for free pro ducts of von Neumann algebras. This computation generalizes earlier work on computation of T. S, and Sd invariants For free product algebras. (C) 1999 Academic Press.