On the Baum–Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg conjecture
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- by Yuki Arano and Adam Skalski
- Proc. Amer. Math. Soc. 149 (2021), 5237-5254
- DOI: https://doi.org/10.1090/proc/15598
- Published electronically: September 21, 2021
Abstract:
We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsion. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced $\mathrm {C}^{*}$-algebra of a countable discrete quantum group $\mathbbl {\Gamma }$ implies that $\mathbbl {\Gamma }$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition.References
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Bibliographic Information
- Yuki Arano
- Affiliation: Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8216, Japan
- MR Author ID: 1186453
- Email: y.arano@math.kyoto-u.ac.jp
- Adam Skalski
- Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland
- MR Author ID: 705797
- ORCID: 0000-0003-1661-8369
- Email: a.skalski@impan.pl
- Received by editor(s): September 15, 2020
- Received by editor(s) in revised form: March 17, 2021
- Published electronically: September 21, 2021
- Additional Notes: The first author was supported by JSPS KAKENHI Grant Number JP18K13424. The second author was partially supported by the National Science Centre (NCN) grant no. 2014/14/E/ST1/00525
- Communicated by: Adrian Ioana
- © Copyright 2021 by Yuki Arano; Adam Skalski
- Journal: Proc. Amer. Math. Soc. 149 (2021), 5237-5254
- MSC (2020): Primary 46L67; Secondary 46L80
- DOI: https://doi.org/10.1090/proc/15598
- MathSciNet review: 4327428