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On locally compact quantum groups whose algebras are factors. (English) Zbl 1121.46056
Summary: We are interested in examples of locally compact quantum groups \((M,\Delta)\) such that both von Neumann algebras, \(M\) and the dual \(\widetilde M\), are factors. There are a lot of known examples such that \((M,\widehat M)\) are respectively of type \((\text{I}_\infty,\text{I}_\infty)\) but there is no example with factors of other types. We construct new examples of type \((\text{I}_\infty,\text{II}_\infty)\), \((\text{II}_\infty,\text{II}_\infty)\) and \((\text{III}_\lambda,\text{III}_\lambda)\) for each \(\lambda\in[0,1]\). We also show that there is no such example with \(M\) or \(\widehat M\) a finite factor.

46L65 Quantizations, deformations for selfadjoint operator algebras
46L35 Classifications of \(C^*\)-algebras
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
Full Text: DOI
[1] Araki, H.; Woods, J., A classification of factors, Publ. res. inst. math. sci. Kyoto univ. ser. A, 4, 51-130, (1968) · Zbl 0206.12901
[2] Baaj, S.; Skandalis, G.; Vaes, S., Non-semi-regular quantum groups coming from number theory, Comm. math. phys., 235, 139-167, (2003) · Zbl 1029.46113
[3] Blackadar, B., The regular representation of the restricted direct product groups, J. funct. anal., 25, 267-274, (1977) · Zbl 0364.22004
[4] Boca, F.P.; Zaharescu, A., Factors of type III and the distribution of prime numbers, Proc. London math. soc. (3), 80, 145-178, (2000) · Zbl 1029.46096
[5] Connes, A., Une classification des facteurs de type III, Ann. sci. école norm. sup. (4), 6, 133-252, (1973) · Zbl 0274.46050
[6] Kustermans, J.; Vaes, S., Locally compact quantum groups, Ann. sci. école norm. sup. (4), 33, 6, 837-934, (2000) · Zbl 1034.46508
[7] Kustermans, J.; Vaes, S., Locally compact quantum groups in the von Neumann algebraic setting, Math. scand., 92, 1, 68-92, (2003) · Zbl 1034.46067
[8] Stratila, S., Modular theory in operator algebras, (1981), Abacus Press Tunbridge Wells, England · Zbl 0504.46043
[9] Vaes, S., The unitary implementation of a locally compact quantum group action, J. funct. anal., 180, 426-480, (2001) · Zbl 1011.46058
[10] Vaes, S.; Vainerman, L., Extensions of locally compact quantum groups and the bicrossed product construction, Adv. math., 175, 1-101, (2003) · Zbl 1034.46068
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