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 |
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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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