# Division Rings and Group von Neumann Algebras

Division Rings and Group von Neumann Algebras Abstract. Let G be a discrete group, let W (G) denote the group von Neumann algebra of (7, and let U (G) denote the set of closed densely defined linear operators affiliated to W(G). If G is torsion free and has a normal free subgroup H such that G/His elementary amenable, then we shall prove that there exists a division ring D such that CG c D c U (G). For G s above, this will establish the integrality of numbers arising from L2-cohomology associated with G. 1991 Mathematics Subject Classification: 22D25, 46L80; 46L10. 1. Introduction Let G be a group and let L2 (G) denote the Hubert space with Hubert basis {g\g e G}. Thus L2 (G) consists of all formal sums X^ G ^g where ag eC and£ tfeG |a 0 | 2 < °o» and has inner product defined by (*.*. heG = « geG geG where - denotes complex conjugation. Let <% denote the set of all closed densely defined linear operators [14, §2.7] considered s acting on the left of I/2 (G), and let <£ denote the subset of * consisting of bounded operators. The adjoint 0* of 0e* satisfies (0w, v) = http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

# Division Rings and Group von Neumann Algebras

Forum Mathematicum, Volume 5 (5) – Jan 1, 1993
16 pages

/lp/de-gruyter/division-rings-and-group-von-neumann-algebras-pN6mcDxsoy
Publisher
de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1993.5.561
Publisher site
See Article on Publisher Site

### Abstract

Abstract. Let G be a discrete group, let W (G) denote the group von Neumann algebra of (7, and let U (G) denote the set of closed densely defined linear operators affiliated to W(G). If G is torsion free and has a normal free subgroup H such that G/His elementary amenable, then we shall prove that there exists a division ring D such that CG c D c U (G). For G s above, this will establish the integrality of numbers arising from L2-cohomology associated with G. 1991 Mathematics Subject Classification: 22D25, 46L80; 46L10. 1. Introduction Let G be a group and let L2 (G) denote the Hubert space with Hubert basis {g\g e G}. Thus L2 (G) consists of all formal sums X^ G ^g where ag eC and£ tfeG |a 0 | 2 < °o» and has inner product defined by (*.*. heG = « geG geG where - denotes complex conjugation. Let <% denote the set of all closed densely defined linear operators [14, §2.7] considered s acting on the left of I/2 (G), and let <£ denote the subset of * consisting of bounded operators. The adjoint 0* of 0e* satisfies (0w, v) =

### Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 1993

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