On the zeroth \(L^2\)-homology of a quantum group (Q2883478)

This article slightly improves the main results of the author in [``\(L^2\)-Betti numbers of coamenable quantum groups'', Münster J. Math. 1, No. 1, 143--179 (2008; Zbl 1195.46073)] and [``\(L^2\)-homology for compact quantum groups'', Math. Scand. 103, No. 1, 111--129 (2008; Zbl 1161.46039)] by removing technical assumptions. It is shown that that the zeroth \(L^2\)-homology of a compact quantum group is non-vanishing if and only if the quantum group is coamenable. For a compact quantum group of Kac type, the zeroth \(L^2\)-Betti number is zero if and only if the quantum group is infinite-dimensional. These results are inspired by older vanishing results for the \(L^2\)-homology of discrete amenable groups.