Going-Down functors and the Künneth formula for crossed products by étale groupoids

Abstract

We study the connection between the Baum-Connes conjecture for an ample groupoid $G$ with coefficient $A$ and the Künneth formula for the K-theory of tensor products by the crossed product $A{⋊}_{r}G$. To do so we develop the machinery of Going-Down functors for ample groupoids. As an application we prove that both the uniform Roe algebra of a coarse space which uniformly embeds into a Hilbert space and the maximal Roe algebra of a space admitting a fibred coarse embedding into a Hilbert space satisfy the Künneth formula. We also provide a stability result for the Künneth formula using controlled K-theory, and apply it to give an example of a space that does not admit a coarse embedding into a Hilbert space, but whose uniform Roe algebra satisfies the Künneth formula. As a by-product of our methods, we also prove a permanence property for the Baum-Connes conjecture with respect to equivariant inductive limits of the coefficient algebra.

arXiv comment32 pages, comments welcome

References in corpus (10)

• Renault's Equivalence Theorem for Groupoid Crossed Products
• Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$-algebras
• A non-amenable groupoid whose maximal and reduced $C^{\ast }$-algebras are the same
• A Going-Down principle for ample groupoids and the Baum-Connes conjecture
• The coarse Baum--Connes conjecture and groupoids II
• Dynamical complexity and controlled operator K-theory
• The maximal coarse Baum-Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space
• Dynamic Asymptotic Dimension: relation to dynamics, topology, coarse geometry, and $C^{\ast }$-algebras
• Groupoid exactness and the weak containment problem
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Cited by in corpus (5)

• A Going-Down principle for ample groupoids and the Baum-Connes conjecture
• Categorical approach to the Baum-Connes conjecture for étale groupoids
• Approximate ideal structures and K-theory
• Coarse Baum-Connes conjecture and rigidity for Roe algebras
• Groupoids decomposition, propagation and K-theory