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Transactions of the American Mathematical Society

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


Authors: Christian Bönicke and Clément Dell’Aiera
Journal: Trans. Amer. Math. Soc. 372 (2019), 8159-8194
MSC (2010): Primary 46L80, 22A22, 19K35
DOI: https://doi.org/10.1090/tran/7913
Published electronically: September 12, 2019
MathSciNet review: 4029694
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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.


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Additional Information

Christian Bönicke
Affiliation: Mathematisches Institut der WWU Münster, Einsteinstrasse 62, 48149 Münster, Germany
Address at time of publication: School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, Scotland
Email: christian.bonicke@glasgow.ac.uk

Clément Dell’Aiera
Affiliation: Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Keller 401A, Honolulu, Hawaii 96822
Email: dellaiera@math.hawaii.edu

DOI: https://doi.org/10.1090/tran/7913
Keywords: K\"unneth formula, groupoid crossed products, Baum--Connes conjecture
Received by editor(s): November 5, 2018
Received by editor(s) in revised form: April 24, 2019, and May 24, 2019
Published electronically: September 12, 2019
Additional Notes: The first author was supported by Deutsche Forschungsgemeinschaft (SFB 878).
Article copyright: © Copyright 2019 American Mathematical Society