Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries
[Actions des groupes quantiques compacts à équivalence monoïdale et applications aux frontières de probabilité]
De Rijdt, An ; Vander Vennet, Nikolas
Annales de l'Institut Fourier, Tome 60 (2010), p. 169-216 / Harvested from Numdam

La notion de l’équivalence monoïdale pour les groupes quantiques compacts a été introduite récemment par Bichon, De Rijdt et Vaes. Dans cet article, nous montrons  : étant donné deux groupes quantiques compacts à équivalence monoïdale, alors il existe une correspondance bijective entre leurs actions. Cette correspondance s’avère être très utile pour obtenir la relation entre les frontières de Poisson et Martin des deux groupes quantiques compacts à équivalence monoïdale. Finalement, nous appliquons ces résultats au calcul des frontières de Poisson des duals associés aux groupes quantiques d’automorphismes.

The notion of monoidal equivalence for compact quantum groups was recently introduced by Bichon, De Rijdt and Vaes. In this paper we prove that there is a natural bijective correspondence between actions of monoidally equivalent quantum groups on unital C * -algebras or on von Neumann algebras. This correspondence turns out to be very useful to obtain the behavior of Poisson and Martin boundaries under monoidal equivalence of quantum groups. Finally, we apply these results to identify the Poisson boundary for the duals of quantum automorphism groups.

Publié le : 2010-01-01
DOI : https://doi.org/10.5802/aif.2520
Classification:  20G42
Mots clés: groupes quantiques, algèbres d’opérateurs, théorie de probabilité
@article{AIF_2010__60_1_169_0,
     author = {De Rijdt, An and Vander Vennet, Nikolas},
     title = {Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries},
     journal = {Annales de l'Institut Fourier},
     volume = {60},
     year = {2010},
     pages = {169-216},
     doi = {10.5802/aif.2520},
     zbl = {pre05703824},
     mrnumber = {2664313},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2010__60_1_169_0}
}
De Rijdt, An; Vander Vennet, Nikolas. Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries. Annales de l'Institut Fourier, Tome 60 (2010) pp. 169-216. doi : 10.5802/aif.2520. http://gdmltest.u-ga.fr/item/AIF_2010__60_1_169_0/

[1] Banica, Teodor Théorie des représentations du groupe quantique compact libre O(n), C. R. Acad. Sci. Paris Sér. I Math., Tome 322 (1996) no. 3, pp. 241-244 | MR 1378260 | Zbl 0862.17010

[2] Banica, Teodor Le groupe quantique compact libre U(n), Comm. Math. Phys., Tome 190 (1997) no. 1, pp. 143-172 | Article | MR 1484551 | Zbl 0906.17009

[3] Banica, Teodor Representations of compact quantum groups and subfactors, J. Reine Angew. Math., Tome 509 (1999), pp. 167-198 | Article | MR 1679171 | Zbl 0957.46038

[4] Banica, Teodor Symmetries of a generic coaction, Math. Ann., Tome 314 (1999) no. 4, pp. 763-780 | Article | MR 1709109 | Zbl 0928.46038

[5] Banica, Teodor Subfactors associated to compact Kac algebras, Integral Equations Operator Theory, Tome 39 (2001) no. 1, pp. 1-14 | Article | MR 1806841 | Zbl 0973.46050

[6] Banica, Teodor Quantum groups and Fuss-Catalan algebras, Comm. Math. Phys., Tome 226 (2002) no. 1, pp. 221-232 | Article | MR 1889999 | Zbl 1034.46062

[7] Bichon, Julien; De Rijdt, An; Vaes, Stefaan Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys., Tome 262 (2006) no. 3, pp. 703-728 | Article | MR 2202309 | Zbl 1122.46046

[8] Boca, Florin P. Ergodic actions of compact matrix pseudogroups on C * -algebras, Astérisque (1995) no. 232, pp. 93-109 (Recent advances in operator algebras (Orléans, 1992)) | MR 1372527 | Zbl 0842.46039

[9] De Rijdt, A.; Vennet, N. Vander Actions of monoidally equivalent compact quantum groups (2006) (Preprint)

[10] Effros, Edward G.; Ruan, Zhong-Jin Discrete quantum groups. I. The Haar measure, Internat. J. Math., Tome 5 (1994) no. 5, pp. 681-723 | Article | MR 1297413 | Zbl 0824.17020

[11] Effros, Edward G.; Ruan, Zhong-Jin Operator spaces, The Clarendon Press Oxford University Press, New York, London Mathematical Society Monographs. New Series, Tome 23 (2000) | MR 1793753 | Zbl 0969.46002

[12] Høegh-Krohn, R.; Landstad, M. B.; Størmer, E. Compact ergodic groups of automorphisms, Ann. of Math. (2), Tome 114 (1981) no. 1, pp. 75-86 | Article | MR 625345 | Zbl 0472.46046

[13] Izumi, Masaki Non-commutative Poisson boundaries and compact quantum group actions, Adv. Math., Tome 169 (2002) no. 1, pp. 1-57 | Article | MR 1916370 | Zbl 1037.46056

[14] Izumi, Masaki; Neshveyev, Sergey; Tuset, Lars Poisson boundary of the dual of SU q (n), Comm. Math. Phys., Tome 262 (2006) no. 2, pp. 505-531 | Article | MR 2200270 | Zbl 1104.58001

[15] Kaimanovich, Vadim A. Boundaries of invariant Markov operators: the identification problem, Ergodic theory of Z d actions (Warwick, 1993–1994), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 228 (1996), pp. 127-176 | MR 1411218 | Zbl 0848.60073

[16] Lance, C. Hilbert C * -modules, a toolkit for operator algebraists (1996) (Leeds) | Zbl 0822.46080

[17] Landstad, Magnus B. Simplicity of crossed products from ergodic actions of compact matrix pseudogroups, Astérisque (1995) no. 232, pp. 111-114 (Appendix to: “Ergodic actions of compact matrix pseudogroups on * -algebras” [Asterisque No. 232 (1995), 93–109; MR1372527 (97d:46075)] by F. P. Boca, Recent advances in operator algebras (Orléans, 1992)) | MR 1372528 | Zbl 0842.46044

[18] Maes, Ann; Van Daele, Alfons Notes on compact quantum groups, Nieuw Arch. Wisk. (4), Tome 16 (1998), pp. 73-112 | MR 1645264 | Zbl 0962.46054

[19] Neshveyev, Sergey; Tuset, Lars The Martin boundary of a discrete quantum group, J. Reine Angew. Math., Tome 568 (2004), pp. 23-70 | Article | MR 2034922 | Zbl 1130.46041

[20] Pinzari, Claudia; Roberts, John E. A duality theorem for ergodic actions of compact quantum groups on C * -algebras, Comm. Math. Phys., Tome 277 (2008) no. 2, pp. 385-421 | Article | MR 2358289 | Zbl 1160.46045

[21] Podleś, Piotr Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups, Comm. Math. Phys., Tome 170 (1995) no. 1, pp. 1-20 | Article | MR 1331688 | Zbl 0853.46074

[22] Sołtan, Piotr M. Quantum Bohr compactification, Illinois J. Math., Tome 49 (2005) no. 4, p. 1245-1270 (electronic) | MR 2210362 | Zbl 1099.46048

[23] Tomatsu, Reiji Amenable discrete quantum groups, J. Math. Soc. Japan, Tome 58 (2006) no. 4, pp. 949-964 | Article | MR 2276175 | Zbl 1129.46061

[24] Tomatsu, Reiji A characterization of right coideals of quotient type and its application to classification of Poisson boundaries, Comm. Math. Phys., Tome 275 (2007) no. 1, pp. 271-296 | Article | MR 2335776 | Zbl 1130.46042

[25] Tomatsu, Reiji Compact quantum ergodic systems, J. Funct. Anal., Tome 254 (2008) no. 1, pp. 1-83 | Article | MR 2375065 | Zbl 1137.46041

[26] Vaes, Stefaan The unitary implementation of a locally compact quantum group action, J. Funct. Anal., Tome 180 (2001) no. 2, pp. 426-480 | Article | MR 1814995 | Zbl 1011.46058

[27] Vaes, Stefaan Strictly outer actions of groups and quantum groups, J. Reine Angew. Math., Tome 578 (2005), pp. 147-184 | Article | MR 2113893 | Zbl 1073.46047

[28] Vaes, Stefaan; Vander Vennet, Nikolas Identification of the Poisson and Martin boundaries of orthogonal discrete quantum groups, J. Inst. Math. Jussieu, Tome 7 (2008) no. 2, pp. 391-412 | Article | MR 2400727 | Zbl 1139.46044

[29] Vaes, Stefaan; Vergnioux, Roland The boundary of universal discrete quantum groups, exactness, and factoriality, Duke Math. J., Tome 140 (2007) no. 1, pp. 35-84 | Article | MR 2355067 | Zbl 1129.46062

[30] Van Daele, A. Discrete quantum groups, J. Algebra, Tome 180 (1996) no. 2, pp. 431-444 | Article | MR 1378538 | Zbl 0864.17012

[31] Van Daele, Alfons; Wang, Shuzhou Universal quantum groups, Internat. J. Math., Tome 7 (1996) no. 2, pp. 255-263 | Article | MR 1382726 | Zbl 0870.17011

[32] Wang, Shuzhou Quantum symmetry groups of finite spaces, Comm. Math. Phys., Tome 195 (1998) no. 1, pp. 195-211 | Article | MR 1637425 | Zbl 1013.17008

[33] Wassermann, Antony Ergodic actions of compact groups on operator algebras. II. Classification of full multiplicity ergodic actions, Canad. J. Math., Tome 40 (1988) no. 6, pp. 1482-1527 | Article | MR 990110 | Zbl 0665.46053

[34] Wassermann, Antony Ergodic actions of compact groups on operator algebras. III. Classification for SU (2), Invent. Math., Tome 93 (1988) no. 2, pp. 309-354 | Article | MR 948104 | Zbl 0692.46058

[35] Wassermann, Antony Ergodic actions of compact groups on operator algebras. I. General theory, Ann. of Math. (2), Tome 130 (1989) no. 2, pp. 273-319 | Article | MR 1014926 | Zbl 0734.46041

[36] Woronowicz, S. L. Compact matrix pseudogroups, Comm. Math. Phys., Tome 111 (1987) no. 4, pp. 613-665 | Article | MR 901157 | Zbl 0627.58034

[37] Woronowicz, S. L. Twisted SU (2) group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci., Tome 23 (1987) no. 1, pp. 117-181 | Article | MR 890482 | Zbl 0676.46050

[38] Woronowicz, S. L. Tannaka-Kreĭn duality for compact matrix pseudogroups. Twisted SU (N) groups, Invent. Math., Tome 93 (1988) no. 1, pp. 35-76 | Article | MR 943923 | Zbl 0664.58044

[39] Woronowicz, S. L. Compact quantum groups, Symétries quantiques (Les Houches, 1995), North-Holland, Amsterdam (1998), pp. 845-884 | MR 1616348 | Zbl 0997.46045