Paschke duality and assembly maps

Abstract

We construct a natural transformation between two versions of G-equivariant K-homology with coefficients in a G-C * -category for a countable discrete group G. Its domain is a coarse geometric K-homology and its target is the usual analytic K-homology. Following classical terminology, we call this transformation the Paschke transformation. We show that under certain finiteness assumptions on a G-space X, the Paschke transformation is an equivalence on X. As an application, we provide a direct comparison of the homotopy theoretic Davis-Lück assembly map with Kasparov's analytic assembly map appearing in the Baum-Connes conjecture. Contents 1 Introduction and statements 2 Constructions with C * -categories 3 G-bornological coarse spaces and KCX G c 4 G-uniform bornological coarse spaces, cones and K G,X C

L‐theory of C∗$C^*$‐algebras

“…As ℍ ≃ ℝ [4] in KK, we have already determined L(ℝ[𝑛]) for 0 ⩽ 𝑛 ⩽ 4. In addition, similarly as in Example 6.3, we have that ΣL(ℍ) ≃ L(ℍ [1]) ≃ L(ℝ [5]) because K 3 (ℝ) = 0. As ℝ[𝑛] ≃ ℝ[𝑛 + 8] in KK by real Bott periodicity, we shall now also calculate the L-groups of the remaining shifts of ℝ, namely ℝ [6] and ℝ [7].…”

“…In [40], it was then observed that the ∞-categorical localization of C * Alg sep at the KK-equivalences is a stably symmetric monoidal ∞-category whose homotopy category is canonically equivalent to the tensor triangulated category KK of Kasparov. This observation has also been taken up in [4] (including extensions of these results to possibly nonseparable 𝐶 * -algebras) in the equivariant case and was used in [5] in a proof of an equivariant form of Paschke duality. Remark 2.7.…”

“…Choosing 𝑀 = KO[−3], we find 𝜋 [0,2] 𝑀 = ℤ [1], so that 5 ℤ because 𝜂 and 𝛽 ℂ are zero on ℤ. However, 𝜋 −2 (𝑀) = 𝜋 1 (KO) ≠ 0, so the claim is shown.…”

“…If 𝐴 is unital, then 𝐴 + ℝ is canonically isomorphic to 𝐴 × ℝ, and likewise in the complex case. (5) The complexification functor is compatible with unitalization, whereas the realification functor is not compatible with unitalization. More precisely the solid diagram commutes, whereas the diagram involving dashed arrows does not.…”

“…The former is a direct consequence of the natural isomorphism between K 𝑛 (−)[ 1 2 ] and L 𝑛 (−)[ 1 2 ] (see also [40]) and the latter follows from Theorem C. □ Remark 3. 5. In what follows, we will crucially use different variants of the ∞-categorical Yoneda lemma.…”