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

Bönicke, C. and Dell'Aiera, C. (2019) Going-down functors and the Künneth formula for crossed products by étale groupoids. Transactions of the American Mathematical Society, 372, pp. 8159-8194. (doi: 10.1090/tran/7913)

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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.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Bonicke, Dr Christian
Authors: Bönicke, C., and Dell'Aiera, C.
College/School:College of Science and Engineering > School of Mathematics and Statistics
Journal Name:Transactions of the American Mathematical Society
Publisher:American Mathematical Society
ISSN:0002-9947
ISSN (Online):1088-6850
Published Online:12 September 2019
Copyright Holders:Copyright © 2019 AMS
First Published:First published in Transactions of the American Mathematical Society 372:8159-8194
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