A-Valued Semicircular Systems

✍ Scribed by Dimitri Shlyakhtenko

• Book ID: 102591518
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 341 KB
• Volume: 166
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3424

✦ Synopsis

To a von Neurnann algebra A and a set of linear maps ' ij : A Ä A, i, j # I such that a [ (' ij ) ij # I : A Ä A B(l 2 (I )) is normal and completely positive, we associate a von Neumann algebra 8(A, '). This von Neumann algebra is generated by A and an A-valued semicircular system X i , i # I, associated to '. In many cases there is a faithful conditional expectation E: 8(A, ') Ä A; if A is tracial, then under certain assumptions on ', 8(A, ') also has a trace. One can think of the construction 8(A, ') 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 8. 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 8(L [0, 1], '). 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 { 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.