We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the ‘weakly contractible’ objects, extending previous work by R. Meyer and R. Nest. We prove the subcategory of weakly contractible objects is complementary to the localizing subcategory of projective objects, which are defined in terms of ‘compactly induced’ algebras with respect to certain proper subgroupoids related to isotropy. The resulting ‘strong’ Baum–Connes conjecture implies the classical one, and its formulation clarifies several permanence properties and other functorial statements. We present multiple applications, including consequences for the Universal Coefficient Theorem, a generalized ‘going-down’ principle, injectivity results for groupoids that are amenable at infinity, the Baum–Connes conjecture for group bundles, and a result about the invariance of K-groups of twisted groupoid $C^*$-algebras under homotopy of twists.

We define and compare two bivariant generalizations of the topological K - group Ktop(G). We consider the Baum-Connes conjecture in this context and study its relation to the usual Baum-Connes conjecture.

Let G be a locally compact group and ρ a non-unitary finite dimensional representation of G. We consider tensor products of ρ by some unitary representations of G in order to define two Banach algebras analogous to the group C*-algebras, C*(G) and C*r(G). We calculate the K-theory of such algebras for a large class of groups satisfying the Baum-Connes conjecture.

We prove that the Bost conjecture with C*-algebra coefficients for locally compact Hausdorff groups passes to open subgroups. We also prove that if a locally compact Hausdorff group acts on a tree, then the Bost conjecture with C*-coefficients is true for the group if and only if it is true for the stabilisers of the vertices.

Given two equivalent locally compact Hausdorff groupoids, We prove that the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C*- coefficients. To show these results, the functoriality of Lafforgue's KK-theory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L1-algebras (with Banach algebra coefficients).