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Division rings and group von Neumann algebras. (English) Zbl 0794.22008

Let \(G\) be a discrete group, let \(W(G)\) denote the group von Neumann algebra of \(G\) and let \(U(G)\) denote the set of closed densely defined linear operators affiliated to \(W(G)\). The author considers in this paper the following conjectures: 1) If \(G\) is torsion free, then there exists a division ring \(D\), such that \(\mathbb{C} G\subset D\subset U(G)\). 2) If \(G\) is torsion free, if \(0\neq\alpha\in \mathbb{C} G\) and \(0\neq\beta\in L^ 2(G)\), then \(\alpha* \beta\neq 0\).
For \(n\in\mathbb{N}\) let \(D_ n(G)\) denote the division closure of \(M_ n(\mathbb{C} G)\) in \(M_ n (U(G))\) and let \(W_ n(G)= M_ n(W(G)) \cap D_ n(G)\). Let \({\mathcal C}\) denote the smallest class of groups which contains all free groups, which is closed under directed unions and which satisfies \(G\in {\mathcal C}\) whenever \(H\vartriangleleft G\), \(H\in {\mathcal C}\) and \(G/H\) is elementary amenable. The following theorem is proved: let \(n\in \mathbb{N}\), let \(G\vartriangleleft P\) be groups and suppose the finite subgroups of \(G\) have bounded order. Write \(l\) for the \(lcm\) of the orders of the finite subgroups of \(G\). Then \(d_ n(G)= M_ n(G)= h^{- 1} D_ n(G)h\) for all \(h\in H\), \(D_ n(G)\) is a semisimple Artinian ring, if \(\alpha\in D_ n(G)\), then there exists \(\beta\in W_ n(G)\) and a nonzero divisor \(\gamma\in W_ n(G)\) such that \(\alpha= \beta\gamma^{-1}\). This theorem shows that the conjecture 1) mentioned above is true for groups \(E\vartriangleleft G\) such that \(G\) is torsion free, \(E\) is free and \(G/E\) is elementary amenable. If we combine the theorem of this paper with author’s theorem 2 in the paper reviewed above [Zbl 0794.22007] we see that the conjecture 2) is true if \(E\vartriangleleft H\vartriangleleft G\) are groups such that \(G\) is torsion free, \(E\) is free and \(H/E\) is elementary amenable and \(G/H\) is right orderable.
Reviewer: J.Ludwig (Metz)

MSC:

22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L10 General theory of von Neumann algebras
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
46L80 \(K\)-theory and operator algebras (including cyclic theory)
16K40 Infinite-dimensional and general division rings

Citations:

Zbl 0794.22007
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