L2-Betti numbers of locally compact groups

Vanishing of $\ell ^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence

“…Petersen [19] extended Lück's dimension function from finite von Neumann algebras to semifinite von Neumann algebras. The dimension function dim µ with respect to (G, µ) is a non-trivial dimension for (algebraic) right L(G)-modules that is additive for short exact sequences of L(G)-modules.…”

𝐿²-Betti numbers of locally compact groups and their cross section equivalence relations

“…This was extended to the case of semifinite von Neumann algebras (N, Tr) in [Pe11, Appendix B]. We give here a more direct approach to the results of [Pe11].…”

“…Following [Pe11], we now define the L 2 -Betti numbers, β n (2) (G), of a lcsc unimodular group G as the Murray-von Neumann dimension of the cohomology groups H n (G, L 2 (G)). Recall that the group von Neumann algebra LG of a lcsc group G is defined as the von Neumann algebra generated by the left regular representation of G on L 2 (G).…”

L2-Betti number of discrete and non-discrete groups

“…An alternative algebraic approach was developed by Farber [9]. After that we discuss Petersen's generalisation to von-Neumann algebras with semifinite traces [24].…”

“…So let us assume that K is compact. Endowing G/K with the pushforward ν of µ, one obtains that β p (G/K, ν) = β p (G, µ) [25,Theorem 3.14]. So we may and will assume that the amenable radical of G is trivial.…”