A generalised Green–Julg theorem for proper groupoids and Banach algebras

Abstract

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) .

kk-Theory for Banach algebras II: Equivariance and Green–Julg type theorems

“…The following definition is a refinement of the corresponding definition in Section 5 of [Par09a], compare also [Laf06] and [Par07].…”

“…The operator T acts on E 0 , canonically, by the continuous extension T 0 of the restriction of T to E c . In [Par13a] it is shown (or rather in [Par07]), B)) and that the map (E, T ) → (E 0 , T 0 ) is an isomorphism on the level of homotopy classes. Now consider the following diagram…”

The Bost conjecture and proper Banach algebras

“…See [Gie82] and also Appendix A.2 of [Par07] for more information on this concept of local convexity. Because many proper G-Banach algebras that appear naturally are locally convex and because there is quite some machinery available to treat the general groupoid case, it seems to be advisable to postpone the non-locally C 0 .Z/-convex case to a later date; in any case, if one is given a possibly non-locally convex G-C 0 .Z/-Banach algebra B, one can always form and study the closely related Gelfand transform G.B/ which is locally C 0 .Z/-convex, compare Section 1.3 of [Par13b].…”

“…This is a first positive result for coefficients which are not C -algebras. The proof does not make use of C -algebraic methods either but rests on a generalised version of the Green-Julg theorem for Banach algebras as presented in [Par13b], which enters in Lemma 3.1:…”

kk-Theory for Banach algebras I: The non-equivariant case

“…pair in the sense of[Par10], see also Definition 1.9 above. The operator T has to be B[0, 1]-linear.…”