Biswarup Das, Matthew Daws, Pekka Salmi, Admissibility Conjecture and Kazhdan's Property (T) for quantum groups, Journal of Functional Analysis, Volume 276, Issue 11, 2019, Pages 3484-3510, ISSN 0022-1236, https://doi.org/10.1016/j.jfa.2018.09.001
Admissibility Conjecture and Kazhdan’s Property (T) for quantum groups
|Author:||Das, Biswarup1,2; Daws, Matthew3; Salmi, Pekka1|
1Department of Mathematical Sciences, University of Oulu, Finland
2Instytut Matematyczny, Uniwersytet Wrocławski, Poland
3Jeremiah Horrocks Institute, University of Central Lancashire, UK
|Persistent link:|| http://urn.fi/urn:nbn:fi-fe202003107787
|Publish Date:|| 2020-11-06
We give a partial solution to a long-standing open problem in the theory of quantum groups, namely we prove that all finite-dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups (Admissibility Conjecture for quantum group representations). We use this to study Kazhdan’s Property (T) for quantum groups with non-trivial scaling group, strengthening and generalising some of the earlier results obtained by Fima, Kyed and Sołtan, Chen and Ng, Daws, Skalski and Viselter, and Brannan and Kerr. Our main results are:
(i) All finite-dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible.
(ii) A generalisation of a theorem of Wang which characterises Property (T) in terms of isolation of finite-dimensional irreducible representations in the spectrum.
(iii) A very short proof of the fact that quantum groups with Property (T) are unimodular.
(iv) A generalisation of a quantum version of a theorem of Bekka–Valette proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of non-existence of almost invariant vectors for weakly mixing representations.
(v) A generalisation of a quantum version of Kerr–Pichot theorem, proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of denseness properties of weakly mixing representations.
Journal of functional analysis
|Pages:||3484 - 3510|
|Type of Publication:||
A1 Journal article – refereed
|Field of Science:||
© 2018 © 2018 Elsevier Inc. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/.