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 [30]. 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.

Amenable and inner amenable actions and approximation properties for crossed products by locally compact groups

“…without the assumption of inner amenable actions but using nuclearity of A ⋊ α G as an A(G)-module in (i) and nuclearity of A ⋊ α ,r G as an A(G)-module in (ii) (see [15,Section 7] for the definition of these notions). Unfortunately, it seems that such a result would require module versions of the deep results linking injectivity, semidiscreteness, and nuclearity, which do not appear to be known in general.…”

“…We have rewritten this paper several times in attempts to account for these works. In particular, Crann [15] and Bearden and Crann [7] also discovered the relationship between amenable actions and module injectivity, and gave another related module property. Our results were obtained independently, but we note that we have been heavily influenced by Crann's earlier work on module injectivity for quantum groups [13,14].…”

“…Since we began this work, there has been a flurry of papers on the topic of amenable actions of locally compact groups. As well as the papers [6,10,11,29,32] mentioned above, which focus on amenable actions on C * -algebras, we have recent work of Crann [15] and Bearden and Crann [7]. We have rewritten this paper several times in attempts to account for these works.…”

Amenable and inner amenable actions and approximation properties for crossed products by locally compact groups

“…We have rewritten this paper several times in attempts to account for these works. In particular, Crann and Bearden-Crann [7,15] also discovered the relationship between amenable actions and module injectivity, and gave another related module property. Our results were obtained independently, but we note that we have been heavily influenced by Crann's earlier work on module injectivity for quantum groups [13,14].…”

“…Since we began this work there has been a flurry of papers on the topic of amenable actions of locally compact groups. As well as the papers [6,10,11,29,32] mentioned above, which focus on amenable actions on C *algebras, we have recent work of Crann [15] and Bearden-Crann [7]. We have rewritten this paper several times in attempts to account for these works.…”