From Quantum Groups to Groups

Kalantar, Mehrdad ; Neufang, Matthias

Canadian journal of mathematics, Tome 65 (2013) no. 5, pp. 1073-1094

Résumé

In this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign to each locally compact quantum group $\mathbb{G}$ a locally compact group $\tilde{\mathbb{G}}$ that is the quantum version of point-masses and is an invariant for the latter. We show that “quantum point-masses” can be identified with several other locally compact groups that can be naturally assigned to the quantum group $\mathbb{G}$ . This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of $\mathbb{G}$ are encoded by $\tilde{\mathbb{G}}$ the latter, despite being a simpler object, can carry very important information about $\mathbb{G}$ .

Détail

DOI : 10.4153/CJM-2012-047-x

Mots-clés : 46L89, locally compact quantum group, locally compact group, von Neumann algebra

Bibliographie

[1] [1] Baaj, S. and Skandalis, G., Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres. Ann. Sci. École Norm. Sup. 26(1993), 425–488. Google Scholar

[2] [2] Baaj, S. and Vaes, S., Double crossed products of locally compact quantum groups. J. Inst. Math. Jussieu 4(2005), 135–173. Google Scholar | DOI

[3] [3] edos, E. and Tuset, L., Amenability and co-amenability for locally compact quantum groups. Internat. J. Math. 14(2003), 865–884. Google Scholar | DOI

[4] [4] Connes, A., Classification of injective factors. Ann. of Math. (2) 104(1976), 73–115. Google Scholar | DOI

[5] [5] de Cannière, J., On the intrinsic group of a Kac algebra. Proc. London Math. Soc. (3) 40(1980), 1–20. Google Scholar | DOI

[6] [6] De Commer, K. and Van Daele, A., Algebraic quantum groups imbedded in C*-algebraic quantum groups. Rocky Mountain J. Math. 40(2010), 1149–1182. Google Scholar | DOI

[7] [7] Effros, E. G. and Ruan, Z.-J., Operator Spaces. London Math. Soc. Monogr. New Series 23, The Clarendon Press, Oxford University Press, New York, 2000. Google Scholar

[8] [8] Enock, M. and Schwartz, J. M., Kac Algebras and Duality of Locally Compact Groups. Springer-Verlag, Berlin, 1992. Google Scholar

[9] [9] Forrest, B. and Runde, V., Amenability and weak amenability of the Fourier algebra. Math. Z. 250(2005), 731–744. Google Scholar | DOI

[10] [10] Haagerup, U., The standard form of von Neumann algebras. Math. Scand. 37(1975), 271–283. Google Scholar

[11] [11] Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis. Vol. I. Structure of Topological Groups, Integration Theory, Group Representations. Second edition. Grundlehren Math.Wiss. 115, Springer-Verlag, Berlin–New York, 1979. Google Scholar

[12] [12] Hu, Z., Neufang, M., and Ruan, Z.-J., Completely bounded multipliers over locally compact quantum groups. Proc. London Math. Soc., to appear. Google Scholar | DOI

[13] [13] Johnson, B. E., Cohomology in Banach algebras. Memoirs of the American Mathematical Society, 127, American Mathematical Society, Providence, RI, 1972. Google Scholar

[14] [14] Junge, M., Neufang, M., Ruan, Z.-J., A representation theorem for locally compact quantum groups. Internat. J. Math. 20(2009), 377–400. Google Scholar | DOI

[15] [15] Kalantar, M., Towards harmonic analysis on locally compact quantum groups: From Groups to Quantum Groups and Back. Ph.D. thesis, Carleton University, Ottawa, 2010. Google Scholar

[16] [16] Kustermans, J., Locally compact quantum groups in the universal setting. Internat. J. Math. 12(2001), 289–338. Google Scholar

[17] [17] Kustermans, J. and Vaes, S., Locally compact quantum groups. Ann. Sci. Ecole Norm. Sup. 33(2000), 837–934. Google Scholar

[18] [18] Kustermans, J., Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92(2003), 68–92. Google Scholar

[19] [19] Leptin, H., Sur l’algèbre de Fourier d’un groupe localement compact. C. R. Acad. Sci. Paris Sér. A–B 266 (1968), A1180–A1182. Google Scholar

[20] [20] Müller, A., Classifying spaces for quantum principal bundles. Comm. Math. Phys. 149(1992), 495–512. Google Scholar | DOI

[21] [21] Renaud, P. F., Centralizers of Fourier algebra of an amenable group. Proc. Amer. Math. Soc. 32(1972), 539–542. Google Scholar

[22] [22] Ruan, Z.-J., The operator amenability of A(G). Amer. J. Math. 117(1995), 1449–1474. Google Scholar | DOI

[23] [23] Runde, V., Lectures on Amenability. Lecture Notes in Math. 1774, Springer-Verlag, Berlin, 2002. Google Scholar

[24] [24] Takesaki, M., A characterization of group algebras as a converse of Tannaka–Stinespring–Tatsuuma duality theorem. Amer. J. Math. 91(1969), 529–564. Google Scholar | DOI

[25] [25] Takesaki, M., Theory of Operator Algebras. II. Encyclopaedia Math. Sci. 125, Operator Algebras and Non-commutative Geometry 6. Springer-Verlag, Berlin, 2003. Google Scholar

[26] [26] Vaes, S., Locally Compact Quantum Groups. Ph.D. thesis, K.U. Leuven, 2000. Google Scholar

[27] [27] G.Wendel, J., Left centralizers and isomorphisms of group algebras. Pacific J. Math. 2(1952), 251–261. Google Scholar

[28] [28] L.Woronowicz, S., Compact matrix pseudogroups. Comm. Math. Phys. 111(1987), 613–665. Google Scholar | DOI

[29] [29] L.Woronowicz, S., Compact quantum groups. In: Sym´etries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 845–884. Google Scholar

[30] [30] L.Woronowicz, S., Quantum E(2) group and its Pontryagin dual. Lett. Math. Phys. 23(1991), 251–263. Google Scholar | DOI

[31] [31] L.Woronowicz, S., Tannaka–Krein duality for compact matrix pseudogroups. Twisted SU(N) groups. Invent. Math. 93(1998), 35–76. Google Scholar | DOI