×

Graphs of quantum groups and \(K\)-amenability. (English) Zbl 1297.46048

Summary: Building on a construction of J.-P. Serre, we associate to any graph of \(C^\ast\)-algebras a maximal and a reduced fundamental \(C^\ast\)-algebra and use this theory to construct the fundamental quantum group of a graph of discrete quantum groups. This construction naturally gives rise to a quantum Bass-Serre tree which can be used to study the \(K\)-theory of the fundamental quantum group. To illustrate the properties of this construction, we prove that, if all the vertex quantum groups are amenable, then the fundamental quantum group is \(K\)-amenable. This generalizes previous results of P. Julg and A. Valette [J. Funct. Anal. 58, 194–215 (1984; Zbl 0559.46030)], R. Vergnioux [ibid. 212, No. 1, 206–221 (2004; Zbl 1064.46064)] and the first author [ibid. 265, No. 4, 507–519 (2013; Zbl 1316.46044)]. Our proof, even for classical groups, is quite different from the original proof of Julg and Valette, which does not seem to extend straightforwardly to the quantum setting.

MSC:

46L65 Quantizations, deformations for selfadjoint operator algebras
46L09 Free products of \(C^*\)-algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baaj, S.; Skandalis, G., \(C^⁎\)-algèbres de Hopf et théorie de Kasparov équivariante, K-Theory, 2, 6, 683-721 (1989) · Zbl 0683.46048
[2] Baum, P.; Connes, A.; Higson, N., Classifying space for proper actions and K-theory of group \(C^⁎\)-algebras, (Contemp. Math., vol. 167 (1994)), 241-292 · Zbl 0830.46061
[3] Bédos, E.; Murphy, G. J.; Tuset, L., Co-amenability of compact quantum groups, J. Geom. Phys., 40, 2, 129-153 (2001) · Zbl 1011.46056
[4] Blackadar, B., K-Theory for Operator Algebras, Math. Sci. Res. Inst. Publ., vol. 5 (1998), Cambridge Univ. Press · Zbl 0913.46054
[5] Boca, F., On the method of constructing irreducible finite index subfactors of Popa, Pacific J. Math., 161, 2, 201-231 (1993) · Zbl 0795.46044
[6] Brown, N. P.; Dykema, K. J.; Jung, K., Free entropy dimension in amalgamated free products, Proc. Lond. Math. Soc., 97, 3, 339-367 (2008), with an appendix by Wolfgang Lück · Zbl 1158.46045
[7] Cuntz, J., K-theoretic amenability for discrete groups, J. Reine Angew. Math., 344, 180-195 (1983) · Zbl 0511.46066
[8] Daws, M.; Fima, P.; Skalski, A.; Stuart, W., The Haagerup property for locally compact quantum groups (2013), preprint
[9] Dykema, K. J., Exactness of reduced amalgamated free product \(C^⁎\)-algebras, Forum Math., 16, 161-180 (2004) · Zbl 1050.46040
[10] Fima, P., K-amenability of HNN extensions of amenable discrete quantum groups, J. Funct. Anal., 265, 4, 507-519 (2013) · Zbl 1316.46044
[11] Fima, P.; Vaes, S., HNN extensions and unique group measure space decomposition of \(II_1\) factors, Trans. Amer. Math. Soc., 364, 5, 2601-2617 (2012) · Zbl 1251.46032
[12] Freslon, A., Propriétés d’approximation pour les groupes quantiques discrets (2013), Université Paris VII, Ph.D. thesis
[13] Germain, E., KK-theory of reduced free-product \(C^⁎\)-algebras, Duke Math. J., 82, 3, 707-724 (1996) · Zbl 0863.46046
[14] Julg, P.; Valette, A., K-theoretic amenability for \(S L_2(Q_p)\), and the action on the associated tree, J. Funct. Anal., 58, 2, 194-215 (1984) · Zbl 0559.46030
[15] Maes, A.; Van Daele, A., Notes on compact quantum groups (1998), preprint · Zbl 0962.46054
[16] Pimsner, M. V., KK-groups of crossed products by groups acting on trees, Invent. Math., 86, 3, 603-634 (1986) · Zbl 0638.46049
[17] Pimsner, M. V.; Voiculescu, D. V., K-groups of reduced crossed products by free groups, J. Operator Theory, 8, 1, 131-156 (1982) · Zbl 0533.46045
[18] Serre, J.-P., Arbres, amalgames, \(SL_2\), Astérisque, 46 (1977) · Zbl 0302.20039
[19] Skandalis, G., Une notion de nucléarité en K-théorie (d’après J. Cuntz), K-Theory, 1, 6, 549-573 (1988) · Zbl 0653.46065
[20] Takesaki, M., Conditional expectations in von Neumann algebras, J. Funct. Anal., 9, 3, 306-321 (1972) · Zbl 0245.46089
[21] Ueda, Y., HNN extensions of von Neumann algebras, J. Funct. Anal., 225, 2, 383-426 (2005) · Zbl 1088.46034
[22] Ueda, Y., Remarks on HNN extensions in operator algebras, Illinois J. Math., 52, 3, 705-725 (2008) · Zbl 1183.46057
[23] Vergnioux, R., K-amenability for amalgamated free products of amenable discrete quantum groups, J. Funct. Anal., 212, 1, 206-221 (2004) · Zbl 1064.46064
[24] Vergnioux, R.; Voigt, C., The K-theory of free quantum groups, Math. Ann., 357, 1, 355-400 (2013) · Zbl 1284.46063
[25] Voiculescu, D. V., Symmetries of some reduced free product \(C^⁎\)-algebras, (Lecture Notes in Math., vol. 1132 (1985)), 556-588
[26] Wang, S., Free products of compact quantum groups, Comm. Math. Phys., 167, 3, 671-692 (1995) · Zbl 0838.46057
[27] Woronowicz, S. L., Compact quantum groups, (Symétries quantiques. Symétries quantiques, Les Houches, 1995 (1998)), 845-884 · Zbl 0997.46045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.