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Amalgamated free product over Cartan subalgebra. (English) Zbl 1030.46085
From the introduction: 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 in the type $$\text{II}_1$$ 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 $$D$$: $(M,E^M_D)= (A,E^A_D)*_D (B,E^B_D),$ where $$E^A_D,E^B_D$$ are unique normal conditional expectations from $$A,B$$ onto $$D$$. In §3 we summarize basic facts on Cartan subalgebras needed for later sections, and in §4 we show the existence of a faithful normal state $$\varphi$$ on $$D$$ satisfying $(A_{ \varphi \circ E^A_D})'\cap M\subseteq A.$ this of course shows that $$M$$ is a factor. In the structure analysis on type III factors the continuous decomposition plays an important role. In §5 we compute the continuous decomposition of the amalgamated free product $$M$$ (in terms of those of $$A$$ and $$B)$$, which enables us to deterinine the flow of weights of $$M$$.
Two appendices are given. In the first appendix an amalgamated free product version of what was shown in Barnett’s paper [L. Barnett, Proc. Am. Math. Soc., 123, 543-553 (1995; Zbl 0808.46088)] is obtained. In the second the modular operator and modular conjugation are determined for an amalgamated free product. In particular, we obtain a commutation theorem whose ordinary free product version (i.e., $$D=\mathbb{C} 1)$$ was pointed out by Voiculescu et al.
Finally we would like to point out that the amalgamated free product of von Neumann algebras over a common Cartan algebra can be captured as the groupoid von Neumann algebra associated with the “free product” of relevant measured equivalence relations.

##### MSC:
 46L54 Free probability and free operator algebras
Zbl 0808.46088
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