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 L1(G)-nuclearity of LUC(G) and L1(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.