Amenability and vanishing of \(L^2\)-Betti numbers: an operator algebraic approach (Q444895)

This article unifies several previous results about vanishing of \(L^2\)-Betti numbers for amenable groups, quantum groups, and groupoids in the context of the definition of \(L^2\)-Betti numbers for von Neumann algebras by \textit{A. Connes} and \textit{D. Shlyakhtenko} [J. Reine Angew. Math. 586, 125--168 (2005; Zbl 1083.46034)]. The setup consists of a tower of algebras \(N\subseteq A\subseteq M\), where \(N\) and \(M\) are finite von Neumann algebras, \(A\) is a dense subalgebra in \(M\), and the inclusion \(N\to M\) is trace-preserving. The case \(N=\mathbb C\) is already interesting.NEWLINENEWLINEThe strong Følner condition introduced here is analogous to the Følner condition for groups. If \(N=\mathbb C\), \(A=\mathbb C[G]\) and \(N = LG\) for a discrete group \(G\), then the strong Følner condition holds if and only if the group \(G\) is amenable. A similar remark holds for discrete quantum groups.NEWLINENEWLINEIt is shown that the inclusion \(A\to M\) is dimension-flat, that is, the von Neumann dimension vanishes for the Tor-modules that are required to vanish for a flat ring inclusion. Furthermore, the strong Følner condition implies that~\(M\) is amenable relative to~\(N\); this is the usual notion of amenability or injectivity if \(N=\mathbb C\).NEWLINENEWLINEThe dimension-flatness implies that \(L^2\)-Betti numbers vanish for amenable groups, quantum groups, and measured groupoids or equivalence relations. Furthermore, \(L^2\)-Betti numbers of Connes-Shlyakhtenko vanish for irrational rotation algebras and UHF-algebras.