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

Abstract 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 [37] . We pursue applications to abstract harmonic analysis, where, for a locally compact quantum group G , we show that L 1 ( G ) -nuclearity of LUC ( G ) and L 1 ( G ) -semi-discreteness of L ∞ ( G ) are both equivalent to co-amenability of G . We establish the equivalence between A ( G ) -injectivity of G ⋉ ¯ M , A ( G ) -semi-discreteness of G ⋉ ¯ M , and amenability of W ⁎ -dynamical systems ( M , G , α ) with M injective. We end with remarks on future directions.