  As is well known, the equivalence between amenability of a locally compact
group $G$ and injectivity of its von Neumann algebra $\mathcal{L}(G)$ does not
hold in general beyond inner amenable groups. In this paper, we show that the
equivalence persists for all locally compact groups if $\mathcal{L}(G)$ is
considered as a $\mathcal{T}(L_2(G))$-module with respect to a natural action.
In fact, we prove an appropriate version of this result for every locally
compact quantum group.
