Zero divisors and group von Neumann algebras
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.
scientific article; zbMATH DE number 4127428