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.References
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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
- 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).
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 8159-8194
- MSC (2010): Primary 46L80, 22A22, 19K35
- DOI: https://doi.org/10.1090/tran/7913
- MathSciNet review: 4029694