Amenability and covariant injectivity of locally compact quantum groups II

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 $L^{\mathrm{\infty }}(\widehat{\mathbb{G}})$ as an operator $L^1(\widehat{\mathbb{G}})$-module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $VN(G)$ is 1-injective as an operator module over the Fourier algebra $A(G)$. As an application, we provide a decomposability result for completely bounded $L^1(\widehat{\mathbb{G}})$-module maps on $L^{\mathrm{\infty }}(\widehat{\mathbb{G}})$, 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–St{\o}rmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.