Morita equivalences and KK-theory for Banach algebras

Abstract

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

Induction for Banach Algebras, Groupoids and KKban

“…T˝1 is linear and functorial, and if T is bounded, then kT˝1k Ä kT k. The following proposition generalises Proposition 4.2 of [Par08] and is proved in [Par07b], Proposition 3.1.59.…”

“…As mentioned in [Par08] and proved in [Par07b], Section 3.7, there is a sufficient condition for the homotopy of KK ban G -cycles, the basic idea being the following: If there is a homomorphism between two cycles, then under certain conditions the mapping cylinder of this homomorphism is a homotopy between the cycles. We formulate these conditions here and refer the reader to [Par07b] for the proofs.…”

“…which satisfy a number of associativity and norm conditions; in addition, the binary operations are assumed to be non-degenerate, i.e., the closed linear span of AE < is all of E < etc. A more rigorous definition can be found in [Par08].…”

“…The right vertical arrow is given by the pullback along the inclusion of A.G;A/ into A.L;A/ in the first component and the pushforward along the Morita equivalence of A.L;B/ and A.G;B/ in the second component (compare Theorem 5.29 of[Par08]). We now show the following theorem:Theorem 5.12 Diagram (7) is commutative.…”

“…See [Par07b], Appendix C.1.2 for the technical details. Note that there is also a canonical equivalence directly between A.G;B/ and A.H;Ind H G B/ as these algebras are contained as full corners in A.L;Ind L G B/, see Section 5.2 of [Par08].…”