The unitary implementation of a measured quantum groupoid action

Enock, Michel

Annales mathématiques Blaise Pascal, Tome 17 (2010) no. 2, pp. 233-302.

Frank Lesieur a introduit une notion de groupoïde quantique mesuré, dans le cadre des algèbres de von Neumann, en s’inspirant des groupes quantiques localement compacts de Kustermans et Vaes (dans la version de cette construction faite dans le cadre des algèbres de von Neumann). Dans un article précédent, l’auteur a introduit les notions d’action, de produit croisé, d’action duale d’un groupoïde quantique mesuré ; un théorème de bidulaité des actions a éte démontré. Cet article continue ce programme : nous démontrons l’existence d’une implémentation standard d’une action, et un théorème de bidulaité pour les poids. Sont ainsi généralisés des résultats qui avaient été démontrés par S. Vaes pour les groupes quantiques localement compacts, et par T. Yamanouchi pour les groupoïdes mesurés.

Mimicking the von Neumann version of Kustermans and Vaes’ locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In a former article, the author had introduced the notions of actions, crossed-product, dual actions of a measured quantum groupoid; a biduality theorem for actions has been proved. This article continues that program: we prove the existence of a standard implementation for an action, and a biduality theorem for weights. We generalize this way results which were proved, for locally compact quantum groups by S. Vaes, and for measured groupoids by T. Yamanouchi.

Classification : 46L55, 46L89
Keywords: Measured quantum groupoids, actions, biduality theorems
Mot clés : Groupoïdes quantiques mesurés, actions, théorèmes de bidualité

[1] Baaj, Saad; Skandalis, Georges Unitaires multiplicatifs et dualité pour les produits croisés de $C^{*}$-algèbres, Ann. Sci. École Norm. Sup. (4), Volume 26 (1993) no. 4, pp. 425-488 | Numdam | MR | Zbl

[2] Baaj, Saad; Skandalis, Georges; Vaes, Stefaan Non-semi-regular quantum groups coming from number theory, Comm. Math. Phys., Volume 235 (2003) no. 1, pp. 139-167 | DOI | MR | Zbl

[3] Baaj, Saad; Vaes, Stefaan Double crossed products of locally compact quantum groups, J. Inst. Math. Jussieu, Volume 4 (2005) no. 1, pp. 135-173 | DOI | MR | Zbl

[4] Blanchard, Etienne Tensor products of $C\left(X\right)$-algebras over $C\left(X\right)$, Astérisque (1995) no. 232, pp. 81-92 Recent advances in operator algebras (Orléans, 1992) | MR | Zbl

[5] Blanchard, Étienne Déformations de $C^{*}$-algèbres de Hopf, Bull. Soc. Math. France, Volume 124 (1996) no. 1, pp. 141-215 | Numdam | MR | Zbl

[6] Böhm, Gabriella; Szlachányi, Kornél Weak $C^{*}$-Hopf algebras: the coassociative symmetry of non-integral dimensions, Quantum groups and quantum spaces (Warsaw, 1995) (Banach Center Publ.), Volume 40, Polish Acad. Sci., Warsaw, 1997, pp. 9-19 | MR | Zbl

[7] Bòhm, Gabriella; Szlachónyi, Korníl A coassociative $C^{*}$-quantum group with nonintegral dimensions, Lett. Math. Phys., Volume 38 (1996) no. 4, pp. 437-456 | DOI | MR | Zbl

[8] Connes, A. On the spatial theory of von Neumann algebras, J. Funct. Anal., Volume 35 (1980) no. 2, pp. 153-164 | DOI | MR | Zbl

[9] Connes, Alain Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994 | MR | Zbl

[10] David, Marie-Claude $C^{*}$-groupoïdes quantiques et inclusions de facteurs: structure symétrique et autodualité, action sur le facteur hyperfini de type ${\mathrm{II}}_1$, J. Operator Theory, Volume 54 (2005) no. 1, pp. 27-68 | MR | Zbl

[11] De Commer, Kenny Monoidal equivalence for locally compact quantum groups, 2008 (mathOA/0804.2405, to appear in J. Operator Theory)

[12] Enock, Michel Produit croisé d’une algèbre de von Neumann par une algèbre de Kac, J. Functional Analysis, Volume 26 (1977) no. 1, pp. 16-47 | DOI | MR | Zbl

[13] Enock, Michel Inclusions irréductibles de facteurs et unitaires multiplicatifs. II, J. Funct. Anal., Volume 154 (1998) no. 1, pp. 67-109 | DOI | MR | Zbl

[14] Enock, Michel Inclusions of von Neumann algebras and quantum groupoïds. III, J. Funct. Anal., Volume 223 (2005) no. 2, pp. 311-364 | DOI | MR | Zbl

[15] Enock, Michel Quantum groupoids of compact type, J. Inst. Math. Jussieu, Volume 4 (2005) no. 1, pp. 29-133 | DOI | MR | Zbl

[16] Enock, Michel Measured quantum groupoids in action, Mém. Soc. Math. Fr. (N.S.) (2008) no. 114, pp. ii+150 pp. (2009) | Numdam | MR | Zbl

[17] Enock, Michel Measured Quantum Groupoids with a central basis, 2008 (mathOA/0808.4052, to be published in J. Operator Theory)

[18] Enock, Michel Outer actions of measured quantum groupoids, 2009 (mathOA/0909.1206)

[19] Enock, Michel; Nest, Ryszard Irreducible inclusions of factors, multiplicative unitaries, and Kac algebras, J. Funct. Anal., Volume 137 (1996) no. 2, pp. 466-543 | DOI | MR | Zbl

[20] Enock, Michel; Schwartz, Jean-Marie Produit croisé d’une algèbre de von Neumann par une algèbre de Kac. II, Publ. Res. Inst. Math. Sci., Volume 16 (1980) no. 1, pp. 189-232 | DOI | MR | Zbl

[21] Enock, Michel; Schwartz, Jean-Marie Kac algebras and duality of locally compact groups, Springer-Verlag, Berlin, 1992 (With a preface by Alain Connes, With a postface by Adrian Ocneanu) | MR | Zbl

[22] Enock, Michel; Vallin, Jean-Michel Inclusions of von Neumann algebras, and quantum groupoids, J. Funct. Anal., Volume 172 (2000) no. 2, pp. 249-300 | DOI | MR | Zbl

[23] Jones, V. F. R. Index for subfactors, Invent. Math., Volume 72 (1983) no. 1, pp. 1-25 | DOI | MR | Zbl

[24] Kustermans, Johan; Vaes, Stefaan Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4), Volume 33 (2000) no. 6, pp. 837-934 | DOI | Numdam | MR | Zbl

[25] Kustermans, Johan; Vaes, Stefaan Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand., Volume 92 (2003) no. 1, pp. 68-92 | Numdam | MR | Zbl

[26] Lesieur, Franck Measured quantum groupoids, Mém. Soc. Math. Fr. (N.S.) (2007) no. 109, pp. iv+158 pp. (2008) | Numdam | MR

[27] Masuda, T.; Nakagami, Y.; Woronowicz, S. L. A $C^{*}$-algebraic framework for quantum groups, Internat. J. Math., Volume 14 (2003) no. 9, pp. 903-1001 | DOI | MR | Zbl

[28] Masuda, Tetsuya; Nakagami, Yoshiomi A von Neumann algebra framework for the duality of the quantum groups, Publ. Res. Inst. Math. Sci., Volume 30 (1994) no. 5, pp. 799-850 | DOI | MR | Zbl

[29] Nikshych, Dmitri; Vainerman, Leonid Algebraic versions of a finite-dimensional quantum groupoid, Hopf algebras and quantum groups (Brussels, 1998) (Lecture Notes in Pure and Appl. Math.), Volume 209, Dekker, New York, 2000, pp. 189-220 | MR | Zbl

[30] Nikshych, Dmitri; Vainerman, Leonid A characterization of depth 2 subfactors of ${\mathrm{II}}_1$ factors, J. Funct. Anal., Volume 171 (2000) no. 2, pp. 278-307 | DOI | MR | Zbl

[31] Nikshych, Dmitri; Vainerman, Leonid Finite quantum groupoids and their applications, New directions in Hopf algebras (Math. Sci. Res. Inst. Publ.), Volume 43, Cambridge Univ. Press, Cambridge, 2002, pp. 211-262 | MR | Zbl

[32] Sauvageot, Jean-Luc Sur le produit tensoriel relatif d’espaces de Hilbert, J. Operator Theory, Volume 9 (1983) no. 2, pp. 237-252 | MR | Zbl

[33] Strătilă, Şerban Modular theory in operator algebras, Editura Academiei Republicii Socialiste România, Bucharest, 1981 (Translated from the Romanian by the author) | MR | Zbl

[34] Szlachányi, Kornél Weak Hopf algebras, Operator algebras and quantum field theory (Rome, 1996), Int. Press, Cambridge, MA, 1997, pp. 621-632 | MR | Zbl

[35] Takesaki, M. Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, 125, Springer-Verlag, Berlin, 2003 (Operator Algebras and Non-commutative Geometry, 6) | MR | Zbl

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

[37] Vaes, Stefaan Strictly outer actions of groups and quantum groups, J. Reine Angew. Math., Volume 578 (2005), pp. 147-184 | DOI | MR | Zbl

[38] Vaes, Stefaan; Vainerman, Leonid Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math., Volume 175 (2003) no. 1, pp. 1-101 | DOI | MR | Zbl

[39] Vallin, Jean-Michel Bimodules de Hopf et poids opératoriels de Haar, J. Operator Theory, Volume 35 (1996) no. 1, pp. 39-65 | MR | Zbl

[40] Vallin, Jean-Michel Unitaire pseudo-multiplicatif associé à un groupoïde. Applications à la moyennabilité, J. Operator Theory, Volume 44 (2000) no. 2, pp. 347-368 | MR | Zbl

[41] Vallin, Jean-Michel Groupoïdes quantiques finis, J. Algebra, Volume 239 (2001) no. 1, pp. 215-261 | DOI | MR | Zbl

[42] Vallin, Jean-Michel Multiplicative partial isometries and finite quantum groupoids, Locally compact quantum groups and groupoids (Strasbourg, 2002) (IRMA Lect. Math. Theor. Phys.), Volume 2, de Gruyter, Berlin, 2003, pp. 189-227 | MR | Zbl

[43] Vallin, Jean-Michel Measured quantum groupoids associated with matched pairs of locally compact groupoids, 2009 (mathOA/0906.5247)

[44] Woronowicz, S. L. Tannaka-Kreĭn duality for compact matrix pseudogroups. Twisted $\mathrm{SU}\left(N\right)$ groups, Invent. Math., Volume 93 (1988) no. 1, pp. 35-76 | DOI | MR | Zbl

[45] Woronowicz, S. L. From multiplicative unitaries to quantum groups, Internat. J. Math., Volume 7 (1996) no. 1, pp. 127-149 | DOI | MR | Zbl

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

[47] Yamanouchi, Takehiko Crossed products by groupoid actions and their smooth flows of weights, Publ. Res. Inst. Math. Sci., Volume 28 (1992) no. 4, pp. 535-578 | DOI | MR | Zbl

[48] Yamanouchi, Takehiko Dual weights on crossed products by groupoid actions, Publ. Res. Inst. Math. Sci., Volume 28 (1992) no. 4, pp. 653-678 | DOI | MR | Zbl

[49] Yamanouchi, Takehiko Duality for actions and coactions of measured groupoids on von Neumann algebras, Mem. Amer. Math. Soc., Volume 101 (1993) no. 484, pp. vi+109 | MR | Zbl

[50] Yamanouchi, Takehiko Canonical extension of actions of locally compact quantum groups, J. Funct. Anal., Volume 201 (2003) no. 2, pp. 522-560 | DOI | MR | Zbl

[51] Yamanouchi, Takehiko Takesaki duality for weights on locally compact quantum group covariant systems, J. Operator Theory, Volume 50 (2003) no. 1, pp. 53-66 | MR | Zbl