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Thoma type results for discrete quantum groups

Teodor Banica

http://orcid.org/0000-0003-4680-0364

Department of Mathematics, University of Cergy-Pontoise, F-95000 Cergy-Pontoise, France

E-mail Address: teo.banica@gmail.com

Corresponding author.

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and

Alexandru Chirvasitu

Department of Mathematics, University of Buffalo, Buffalo, NY 14260-2900, USA

E-mail Address: achirvas@buffalo.edu

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https://doi.org/10.1142/S0129167X17501038Cited by:2 (Source: Crossref)

Abstract

Thoma's theorem states that a group algebra $C^{*} (Γ)$ is of type I if and only if $Γ$ is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually abelianity conditions on a discrete quantum group $Γ$ , we have a stationary model of type $π : C^{*} (Γ) \to M_{F} (C (L))$ , with $F$ being a finite quantum group, and with $L$ being a compact group. We discuss then some refinements of these results in the quantum permutation group case, $\hat{Γ} \subset S_{N}^{+}$ , by restricting the attention to the matrix models which are quasi-flat, in the sense that the images of the standard coordinates, known to be projections, have rank $\leq 1$ .

Keywords:

AMSC: 46L54, 60B15

References

1. N. Andruskiewitsch and J. Devoto, Extensions of Hopf algebras, St. Petersburg Math. J. 7 (1996) 17–52. Google Scholar
2. T. Banica, Quantum groups from stationary matrix models, Colloq. Math. 148 (2017) 247–267. Crossref, Web of Science, Google Scholar
3. T. Banica and J. Bichon, Quantum groups acting on $4$ points, J. Reine Angew. Math. 626 (2009) 74–114. Web of Science, Google Scholar
4. T. Banica and J. Bichon, Matrix models for noncommutative algebraic manifolds, J. Lond. Math. Soc. 95 (2017) 519–540. Crossref, Web of Science, Google Scholar
5. T. Banica and J. Bichon, Complex analogues of the half-classical geometry, Münster J. Math. 10 (2017) 457–483. Google Scholar
6. T. Banica and B. Collins, Integration over the Pauli quantum group, J. Geom. Phys. 58 (2008) 942–961. Crossref, Web of Science, Google Scholar
7. T. Banica and A. Freslon, Modelling questions for quantum permutations, preprint (2017), arXiv:1704.00290. Google Scholar
8. T. Banica and I. Nechita, Flat matrix models for quantum permutation groups, Adv. Appl. Math. 83 (2017) 24–46. Crossref, Web of Science, Google Scholar
9. E. Bédos, G. J. Murphy and L. Tuset, Co-amenability of compact quantum groups, J. Geom. Phys. 40 (2001) 129–153. Crossref, Web of Science, Google Scholar
10. J. Bichon, Algebraic quantum permutation groups, Asian-Eur. J. Math. 1 (2008) 1–13. Link, Google Scholar
11. J. Bichon, Half-liberated real spheres and their subspaces, Colloq. Math. 144 (2016) 273–287. Web of Science, Google Scholar
12. J. Bichon and M. Dubois-Violette, Half-commutative orthogonal Hopf algebras, Pacific J. Math. 263 (2013) 13–28. Crossref, Web of Science, Google Scholar
13. K. Bragiel, The twisted $S U (N)$ group. On the $C^{*}$ -algebra $C (S_{μ} U (N))$ , Lett. Math. Phys. 20 (1990) 251–257. Crossref, Web of Science, Google Scholar
14. M. Brannan, B. Collins and R. Vergnioux, The Connes embedding property for quantum group von Neumann algebras, Trans. Amer. Math. Soc. 369 (2017) 3799–3819. Crossref, Web of Science, Google Scholar
15. A. Chirvasitu, Residually finite quantum group algebras, J. Funct. Anal. 268 (2015) 3508–3533. Crossref, Web of Science, Google Scholar
16. A. Chirvasitu, S. O. Hoche and P. Kasprzak, Fundamental isomorphism theorems for quantum groups, preprint (2016), arXiv:1610.04105. Google Scholar
17. L. S. Cirio, A. D'Andrea, C. Pinzari and S. Rossi, Connected components of compact matrix quantum groups and finiteness conditions, J. Funct. Anal. 267 (2014) 3154–3204. Crossref, Web of Science, Google Scholar
18. D. Enders, A characterization of semiprojectivity for subhomogeneous $C^{*}$ -algebras, Doc. Math. 21 (2016) 987–1049. Crossref, Web of Science, Google Scholar
19. M. Enock and L. Vainerman, Deformation of a Kac algebra by an abelian subgroup, Comm. Math. Phys. 178 (1996) 571–596. Crossref, Web of Science, Google Scholar
20. P. Fima, On locally compact quantum groups whose algebras are factors, J. Funct. Anal. 244 (2007) 78–94. Crossref, Web of Science, Google Scholar
21. U. Franz and A. Skalski, On idempotent states on quantum groups, J. Algebra 322 (2009) 1774–1802. Crossref, Web of Science, Google Scholar
22. P. Józiak, Remark on Hopf images in quantum permutation groups $S_{n}^{+}$ , preprint (2016), arXiv:1611.09211. Google Scholar
23. E. Kaniuth, Der Typ der regulären Darstellung diskreter Gruppen, Math. Ann. 182 (1969) 334–339. Crossref, Web of Science, Google Scholar
24. H. F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J. 5 (1981) 675–692. Crossref, Web of Science, Google Scholar
25. S. Montgomery, Hopf Algebras and Their Actions on Rings (American Mathematical Society, 1993). Crossref, Google Scholar
26. F. J. Murray and J. von Neumann, On rings of operators, Ann. of Math. 37 (1936) 116–229. Crossref, Google Scholar
27. S. Raum and M. Weber, The full classification of orthogonal easy quantum groups, Comm. Math. Phys. 341 (2016) 751–779. Crossref, Web of Science, Google Scholar
28. H. J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990) 167–195. Crossref, Web of Science, Google Scholar
29. M. Smith, Regular representations of discrete groups, J. Funct. Anal. 11 (1972) 401–406. Crossref, Google Scholar
30. M. Takeuchi, Relative Hopf modules — equivalences and freeness criteria, J. Algebra 60 (1979) 452–471. Crossref, Web of Science, Google Scholar
31. P. Tarrago and M. Weber, Unitary easy quantum groups: The free case and the group case, preprint (2015). Google Scholar
32. E. Thoma, Über unitäre Darstellungen abzählbarer, diskreter Gruppen, Math. Ann. 153 (1964) 111–138. Crossref, Web of Science, Google Scholar
33. R. Vergnioux and C. Voigt, The K-theory of free quantum groups, Math. Ann. 357 (2013) 355–400. Crossref, Web of Science, Google Scholar
34. J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50 (1949) 401–485. Crossref, Web of Science, Google Scholar
35. S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995) 671–692. Crossref, Web of Science, Google Scholar
36. S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998) 195–211. Crossref, Web of Science, Google Scholar
37. S. Wang, $L_{p}$ -improving convolution operators on finite quantum groups, Indiana Univ. Math. J. 65 (2016) 1609–1637. Crossref, Web of Science, Google Scholar
38. S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987) 613–665. Crossref, Web of Science, Google Scholar
39. S. L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988) 35–76. Crossref, Web of Science, Google Scholar
40. S. L. Woronowicz, Compact quantum groups, in Symétries Quantiques (North-Holland, 1998), pp. 845–884. Google Scholar