Going-Down functors and the K\"unneth formula for crossed products by \'etale groupoids

TL;DR

This paper explores the relationship between the Baum-Connes conjecture and the K"unneth formula for crossed products by ample groupoids, developing new tools and proving results for Roe algebras and stability under inductive limits.

Contribution

It introduces Going-Down functors for ample groupoids and proves the K"unneth formula for certain Roe algebras, also establishing stability results and permanence properties.

Findings

The K"unneth formula holds for uniform and maximal Roe algebras of spaces with specific embeddings.

A stability result for the K"unneth formula using controlled K-theory is established.

A permanence property for the Baum-Connes conjecture under inductive limits is proved.

Abstract

We study the connection between the Baum-Connes conjecture for an ample groupoid $G$ with coefficient $A$ and the K\"unneth formula for the K-theory of tensor products by the crossed product $A\rtimes_r G$. 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 K\"unneth formula. We also provide a stability result for the K\"unneth 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 K\"unneth 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.