Amalgamated Free Product over Cartan Subalgebra
نویسندگان
چکیده
Amalgamated free products of von Neumann algebras were first used by S. Popa ([26]) to construct an irreducible inclusion of (non-AFD) type II1 factors with an arbitrary (admissible) Jones index. Further investigation in this direction was made by K. Dykema ([10]) and F. Rădulescu ([27, 29]) based on Voiculescu’s powerful machine ([40, 41, 44]), and F. Boca ([4]) discussed the Haagerup approximation property, where only finite von Neumann algebras were dealt with. On the other hand, type III factors arising as free products (over C) were studied by L. Barnett ([3]), K. Dykema ([9, 11]), F. Rădulescu ([28]), and very recently by D. Shlyakhtenko ([33]). However, amalgamated free products in the type III setting have never been seriously investigated so far. The main purpose of the paper is to take a first step towards investigation on amalgamated free products in the type III setting. A construction of amalgamated free products of arbitrary von Neumann algebras has never been (at least explicitly) given in the literature (see [29, 44] in the type II1 case), and hence we present such a construction in §2. Our construction requires (faithful) normal conditional expectations onto a common subalgebra, and the concept of bimodules is useful. We mainly study the amalgamated free product of non-type I factors A,B over their common Cartan subalgebra D:
منابع مشابه
N ov 2 00 2 AMALGAMATED FREE PRODUCT OVER CARTAN SUBALGEBRA , II
Let A ⊇ D ⊆ B be two von Neumann algebras together with a common Cartan subalgebra. Then the amalgamated free product M = A ∗D B with respect to the unique conditional expectations from A, B onto D can be considered. In our previous paper [U1], the questions of its factoriality and type classification were discussed in detail, which will be reviewed in §4. The main purpose of the paper is to gi...
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