Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of ${{L}^{\infty }}\left( \widehat{\mathbb{G}} \right)$ as an operator ${{L}^{1}}\left( \widehat{\mathbb{G}} \right)$-module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $\text{VN}\left( G \right)$ is 1-injective as an operator module over the Fourier algebra $A\left( G \right)$. As an application, we provide a decomposability result for completely bounded ${{L}^{1}}\left( \widehat{\mathbb{G}} \right)$-module maps on ${{L}^{\infty }}\left( \widehat{\mathbb{G}} \right)$, and give a simplified proof that amenable discrete quantum groups have co-amenable compact duals, which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

We prove that if ${\it\rho}$ is an irreducible positive definite function in the Fourier–Stieltjes algebra $B(G)$ of a locally compact group $G$ with $\Vert {\it\rho}\Vert _{B(G)}=1$, then the iterated powers $({\it\rho}^{n})$ as a sequence of unital completely positive maps on the group $C^{\ast }$-algebra converge to zero in the strong operator topology.