In this paper we introduce and study new notions of amenability for actions of locally compact groups on C ˚-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and consequences analogous to those known in the discrete case.One of our main results generalises a theorem of Matsumura: we show that for an action of an exact locally compact group G on a locally compact space X the full and reduced crossed products C 0 pXq¸max G and C 0 pXq¸r ed G coincide if and only if the action of G on X is measurewise amenable. We also show that the analogue of this theorem does not hold for actions on noncommutative C ˚-algebras.

Abstract. C * -algebras form a 2-category with * -homomorphisms or correspondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions, and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby-Smith twisted actions and equivalence of such actions, covariant representations, and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green and Busby-Smith.The Packer-Raeburn Stabilisation Trick implies that all Busby-Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalise this to actions of strict 2-groupoids.

We study general properties of exotic crossed-product functors and characterise those which extend to functors on equivariant C * -algebra categories based on correspondences. We show that every such functor allows the construction of a descent in KK-theory and we use this to show that all crossed products by correspondence functors of K-amenable groups are KK-equivalent. We also show that for second countable groups the minimal exact Morita compatible crossed-product functor used in the new formulation of the Baum-Connes conjecture by Baum, Guentner and Willett ([4]) extends to correspondences when restricted to separable G-C * -algebras. It therefore allows a descent in KK-theory for separable systems.2010 Mathematics Subject Classification. 46L55, 46L08, 46L80.

Given a C * -dynamical system (A, G, α), we say that A is a weakly proper X ⋊ G-algebra if there exists a proper G-space X together with a nondegenerate G-equivariant * -homomorphism φ : C 0 (X) → M(A). Weakly proper G-algebras form a large subclass of the class of proper G-algebras in the sense of Rieffel. In this paper we show that weakly proper X ⋊ G-algebras allow the construction of full fixed-point algebras A G,α u corresponding to the full crossed product A ⋊α G, thus solving, in this setting, a problem stated by Rieffel in his 1988's original article on proper actions. As an application we obtain a general Landstad duality result for arbitrary coactions together with a new and functorial construction of maximalizations of coactions.The same methods also allow the construction of exotic generalized fixedpoint algebras associated to crossed-product norms lying between the reduced and universal ones. Using these, we give complete answers to some questions on duality theory for exotic crossed products recently raised by Kaliszewski, Landstad and Quigg. c

We define inverse semigroup actions on topological groupoids by partial equivalences. From such actions, we construct saturated Fell bundles over inverse semigroups and non-Hausdorff \'etale groupoids. We interpret these as actions on C*-algebras by Hilbert bimodules and describe the section algebras of these Fell bundles. Our constructions give saturated Fell bundles over non-Hausdorff \'etale groupoids that model actions on locally Hausdorff spaces. We show that these Fell bundles are usually not Morita equivalent to an action by automorphisms. That is, the Packer-Raeburn Stabilisation Trick does not generalise to non-Hausdorff groupoids.Comment: 59 pages. Proofreading with minor changes. Version accepted for publication at Rocky Mountain Journa