Amalgamated free product over Cartan subalgebra.

*(English)*Zbl 1030.46085From 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.

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 |