Exotic crossed products and the Baum–Connes conjecture

Abstract

We study general properties of exotic crossed-product functors and characterise those which extend to functors on equivariant C * -algebra categories based on correspondences. We show that every such functor allows the construction of a descent in KK-theory and we use this to show that all crossed products by correspondence functors of K-amenable groups are KK-equivalent. We also show that for second countable groups the minimal exact Morita compatible crossed-product functor used in the new formulation of the Baum-Connes conjecture by Baum, Guentner and Willett ([4]) extends to correspondences when restricted to separable G-C * -algebras. It therefore allows a descent in KK-theory for separable systems.2010 Mathematics Subject Classification. 46L55, 46L08, 46L80.

Exact large ideals of 𝐵(𝐺) are downward directed

“…[BEWa,Remark 3.4] says that the ideal property holds for every crossed-product functor coming from a large ideal. This also follows from the following lemma.…”

“…[BEWa,Subsection 9.2 Question (1)] asks whether, for every exact group G and all p ∈ [2, ∞), the crossed-product functor ⋊ Ep is exact, where E p is the weak*-closure of B(G) ∩ L p (G) (which should be changed to span{P (G) ∩ L p (G)}, as in the discussion preceding Lemma 3.5 of the current paper and in [BEWb, Proposition 2.13]). We know that if G is a free group F n with n > 1, then for 2 ≤ p < ∞ the coaction functor τ Ep is not exact.…”

“…In [KLQa] we speculated that the proof would require a "somewhat more elaborate version of Morita compatibility", and that it would "perhaps resemble the property that Buss, Echterhoff, and Willett call correspondence functoriality (see [BEWa,Theorem 4.9])". It transpires that we ended up doing something slightly different: rather than change our definition of Morita compatibility, we instead combine it with another concept from [BEWa], namely the ideal property.…”

Tensor-Product Coaction Functors

“…For technical reasons, our techniques currently only apply to discrete groups, so from now on we suppose that the group G is discrete. 1 Fix an action (C, γ) of G. Both papers [BGW16,BEW18] require C to be unital. For every action (B, α) of G, first form the diagonal action α ⊗ γ of G on the maximal tensor product B ⊗ max C. The embedding b → b⊗1 from B to B ⊗ max C is G-equivariant, and its crossed product is a homomorphism…”

Exotic Crossed Products

“…In fact, Baum, Guentner, and Willett show in [3] that several of the known counterexamples for the original conjecture satisfy the new conjecture and, so far, there are no known counterexamples for the reformulated conjecture. Motivated by the above described developments, the authors of this paper started in [9] a more systematic study of various functorial properties of exotic crossedproduct functors (A, α) → A ⋊ α,µ G. Recall first that a correspondence between two C*-algebras A and B consists of a Hilbert B-module E B equipped with a left action Φ : A → L(E B ) of A on E B . In case of the universal or reduced crossedproducts ⋊ u or ⋊ r , respectively, it has been known for a long time that they are (1) In § §3-5 we give a survey of the general theory of exotic crossed products and, in particular, the results obtained in [9] in which we give characterisations of crossed products which enjoy strong functorial properties as described above.…”