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Vincent Lafforgue's bivariant K-theory for Banach algebras is invariant in the second variable under a rather general notion of Morita equivalence. In particular, the ordinary topological K-theory for Banach algebras is invariant under Morita equivalences. G on the right and that Morita equivalences give isomorphisms; this implies the invariance of KK ban G in the second variable under Morita equivalences. In the final section ( § 6), we indicate how our results can be generalized from group actions to actions of groupoids as was carried out in [7]. NotationAll Banach spaces and Banach algebras that appear in this article are supposed to be complex. If E is a Banach space, then we write E[0, 1] for the Banach space C([0, 1], E) Morita equivalences and KK-theory for Banach algebras 567with the sup-norm. If E carries some extra structure (for example if E is a Banach algebra), then we equip E[0, 1] with the pointwise structure (so that E[0, 1] is a Banach algebra, too). of these 'rank one' operators is denoted by K B (E, F ), and the elements of K B (E, F ) are called compact. Again, we write K B (E) instead of K B (E, E), and the subscript B is often omitted.Linear and compact operators between Banach pairs generalize adjointable and compact operators between Hilbert modules, respectively. They are the key ingredient in the definition of cycles for KK ban . But there is also a second type of morphisms between Banach pairs which generalizes homomorphisms with coefficient maps between Hilbert modules over possibly different C * -algebras.Definition 1.1 (concurrent homomorphism of pairs). Let B, B be Banach algebras, let E be a B-pair and E a B -pair. A concurrent homomorphism Ψ from E to E is a pair Ψ = (Ψ < , Ψ > ) together with a so-called coefficient map ψ of Ψ , where Ψ < : E < → E < and Ψ > : E > → E > are C-linear and contractive maps and ψ : B → B

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 Ccoefficients. 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 L 1 -algebras (with Banach algebra coefficients).

The Green-Julg theorem states that K G 0 (B) ∼ = K 0 (L 1 (G, B)) for every compact group G and every G-C * -algebra B. We formulate a generalisation of this result to proper groupoids and Banach algebras and deduce that the Bost assembly map is surjective for proper Banach algebras. Definition of RKK banG (C 0 (X); A, B) Definition 1.5. Let A and B be G-C 0 (X)-Banach algebras. Then the class E ban G (C 0 (X); A, B) is defined to be the class of pairs (E, T ) such that E is a non-degenerate graded G-C 0 (X)-Banach A-B-pair and, if we forget the C 0 (X)-structure, the pair (E, T ) is an element of E ban G (A, B). Note that T in the definition is automatically C 0 (X)-linear because E is non-degenerate.The constructions one usually performs with KK ban -cycles are obviously compatible with the additional C 0 (X)-structure, so we can form the sum of KK ban -cycles and take their pushout along homomorphisms of G-C 0 (X)-Banach algebras. We also have a C 0 (X)-linear notion of morphisms of KK ban -cycles, giving us a C 0 (X)-linear version of isomorphisms of KK ban -cycles. Hence also the notion of homotopy makes sense in the C 0 (X)-setting so we can formulate the following definition:Definition 1.6. The class of all homotopy classes of elements of E ban G (C 0 (X); A, B) is denoted by RKK ban G (C 0 (X); A, B). The sum of cycles induces a law of composition on RKK ban G (C 0 (X); A, B) making it an abelian group.The fact that the composition on RKK ban G (C 0 (X); A, B) has inverses can be proved just as in the case without the C 0 (X)-structure, i.e., Lemme 1.2.5 of [Laf02] and its proof are compatible with the additional C 0 (X)-module action. There is an obvious forgetful group homomorphism RKK ban G (C 0 (X); A, B) → KK ban G (A, B) .

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.