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 quantitative analogues finite-dimensionality for operator modules over completely contractive Banach algebras. With these we begin the development a local theory by introducing lifting property, nuclearity, semi-discreteness. For large class prove that property is equivalent to flatness, generalizing space result Kye Ruan [37]. pursue applications abstract harmonic analysis, where, locally compact quantum group G, show L1(G)-nuclearity LUC(G) L1(G)-semi-discreteness L∞(G) are both co-amenability G. establish equivalence between A(G)-injectivity G⋉¯M, A(G)-semi-discreteness amenability W⁎-dynamical systems (M,G,α) with M injective. end remarks on future directions.