Finite presentation, the local lifting property, and local approximation properties of operator modules

We introduce notions of finite presentation and co-exactness which serve as qualitative and quantitative analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we begin the development of a local theory of operator modules by introducing analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator modules we prove that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan. We pursue applications to abstract harmonic analysis, where, for a locally compact quantum group $\mathbb{G}$, we show that $L^1(\mathbb{G})$-nuclearity of $\mathrm{LUC}(\mathbb{G})$ and $L^1(\mathbb{G})$-semi-discreteness of $L^\infty(\mathbb{G})$ are both equivalent to co-amenability of $\mathbb{G}$. We establish the equivalence between $A(G)$-injectivity of $G\bar{\ltimes}M$, $A(G)$-semi-discreteness of $G\bar{\ltimes} M$, and amenability of $W^*$-dynamical systems $(M,G,\alpha)$ with $M$ injective. We end with remarks on future directions.