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

Bönicke, Christian; Dell’Aiera, Clément

doi:10.1090/tran/7913

Going-down functors and the Künneth formula for crossed products by étale groupoids
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by Christian Bönicke and Clément Dell’Aiera PDF

Trans. Amer. Math. Soc. 372 (2019), 8159-8194 Request permission

Abstract:

We study the connection between the Baum–Connes conjecture for an ample groupoid $G$ with coefficient $A$ and the Künneth formula for the ${\mathrm K}$-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 in a Hilbert space and the maximal Roe algebra of a space admitting a fibered coarse embedding in a Hilbert space satisfy the Künneth formula. Additionally, we give an example of a space that does not admit a coarse embedding in a Hilbert space, but whose uniform Roe algebra satisfies the Künneth formula and provides a stability result for the Künneth formula using controlled ${\mathrm K}$-theory. As a byproduct of our methods, we also prove a permanence property for the Baum–Connes conjecture with respect to equivariant inductive limits of the coefficient algebra.

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