Zero divisors and group von Neumann algebras
type: Item
instance of: scholarly article
title: Zero divisors and group von Neumann algebras
zbMATH Open document ID: 0688.16011
published in: Pacific Journal of Mathematics
publication date: 1991-01-01T00:00:00Z
review text: Let G be a discrete group, let \(L^ 2(G)\) denote the Hilbert space with Hilbert basis the elements of G, and let W(G) denote the group von Neumann algebra of G. The class of elementary amenable groups is the smallest class of groups which contains all abelian and all finite groups, is extension closed, and is closed under directed unions. It is also closed under subgroups and quotient groups, and contains all solvable by finite groups. On the other hand any group which contains a subgroup isomorphic to a nonabelian free group is not elementary amenable. In this paper it is proved that if G is a torsion free elementary amenable group, \(\alpha\in {\mathbb{C}}G\), \(\beta \in L^ 2(G)\), and \(\alpha\),\(\beta\neq 0\), then \(\alpha\) \(\beta\neq 0\). The proof of this relies heavily on the affirmative answer to the zero divisor conjecture for elementary amenable groups which was recorded in Theorem 1.4 of \textit{P. H. Kropholler}, \textit{P. A. Linnell} and \textit{J. A. Moody} [Proc. Am. Math. Soc. 104, 675-684 (1988; Zbl 0691.16013)], and was made possible by Moody's remarkable induction theorem [\textit{J. A. Moody}, J. Algebra 122, 1-14 (1989)]. Also important in the proof are the techniques of \textit{M. K. Smith} [Proc. Am. Math. Soc. 91, 529-531 (1984; Zbl 0607.16009)]. The above is generalized to the case when G is an elementary amenable group (not necessarily torsion free) whose finite subgroups have bounded order. If n is a positive integer, then it is shown that \(M_ n({\mathbb{C}}G)\) and \(M_ n(W(G))\) have classical quotient rings Q and U respectively such that \(Q\subseteq U\). Other related results are also proved.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 4127428
zbMATH Keywords: Hilbert space
zbMATH Keywords: group von Neumann algebra
zbMATH Keywords: elementary amenable groups
zbMATH Keywords: solvable by finite groups
zbMATH Keywords: zero divisor conjecture
zbMATH Keywords: induction theorem
zbMATH Keywords: classical quotient rings
author: Peter A. Linnell
MaRDI profile type: Publication
full work available at URL: pjm.1991.149.349
OpenAlex ID: W2020366789
DOI: 10.2140/PJM.1991.149.349
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description: scientific article; zbMATH DE number 4127428