Actions and coactions of finite quantum groupoids on von Neumann algebras, extensions of the matched pair procedure. (English) Zbl 1128.46026

Summary: Actions and coactions of finite \(C^{*}\)-quantum groupoids are studied in an operator algebras context. In particular, we prove a double crossed product theorem, and the existence of a universal von Neumann algebra on which any finite groupoid acts outerly. We give two actually different extensions of the matched pairs procedure. In [Publ.Mat.Urug.10, 11–51 (2005; Zbl 1092.16021)], N.Andruskiewitsch and S.Natale defined, for any matched pair of groupoids, two \(C^{*}\)-quantum groupoids in duality; we give here an interpretation of them in terms of crossed products of groupoids using a single multiplicative partial isometry which gives a complete description of these structures. The second extension deals only with groups to define another type of finite \(C^{*}\)-quantum groupoids.


46L65 Quantizations, deformations for selfadjoint operator algebras
22A22 Topological groupoids (including differentiable and Lie groupoids)


Zbl 1092.16021
Full Text: DOI arXiv


[1] Aguiar, M.; Andruskiewitsch, N., Representations of matched pairs of groupoids and applications to weak Hopf algebras, (Contemp. Math., vol. 376 (2005)) · Zbl 1100.16032
[2] Andruskiewitsch, N.; Natale, S., Double categories and quantum groupoids, Publ. Mat. Urug., 10, 11-51 (2005) · Zbl 1092.16021
[3] Baaj, S.; Skandalis, G., Unitaires multiplicatifs et dualité pour les produits croisés de \(C^\ast \)-algèbres, Ann. Sci. École Norm. Sup., 26, 425-488 (1993) · Zbl 0804.46078
[4] Baaj, S.; Blanchard, E.; Skandalis, G., Unitaires multiplicatifs en dimension finie et leurs sous-objets, Ann. Inst. Fourier, 49, 1305-1344 (1999) · Zbl 0938.46050
[5] Bisch, D.; Haagerup, U., Composition of subfactors: New examples of infinite depth subfactors, Ann. Sci. École Norm. Sup. (4), 29, 3, 329-383 (1996) · Zbl 0853.46062
[6] Böhm, G.; Szlachányi, K., Weak \(C^\ast \)-Hopf algebras: The coassociative symmetry of non-integral dimensions, (Quantum Groups and Quantum Spaces. Quantum Groups and Quantum Spaces, Banach Center Publications, vol. 40 (1997)), 9-19 · Zbl 0894.16018
[7] Böhm, G.; Szlachányi, K.; Nill, F., Weak Hopf algebras I. Integral theory and \(C^\ast \)-structure, J. Algebra, 221, 385-438 (1999) · Zbl 0949.16037
[8] Enock, M., Produit croisé d’une algèbre de von Neumann par une algèbre de Kac, J. Funct. Anal., 26, 16-47 (1977) · Zbl 0366.46053
[9] Enock, M., Inclusions of von Neumann algebras and quantum groupoids III, J. Funct. Anal., 223, 311-364 (2005) · Zbl 1088.46036
[10] Enock, M.; Vallin, J. M., Inclusions of von Neumann algebras and quantum groupoids, J. Funct. Anal., 172, 249-300 (2000) · Zbl 0974.46055
[11] Gardiner, C. F., Algebraic Structures (1986), Ellis Horwood Limited, John Wiley and Sons · Zbl 0595.20001
[12] F.M. Goodman, P. de la Harpe, V.F.R. Jones, Coxeter graphs and towers of algebras, Math. Soc. Res. Inst. Publ. 14; F.M. Goodman, P. de la Harpe, V.F.R. Jones, Coxeter graphs and towers of algebras, Math. Soc. Res. Inst. Publ. 14 · Zbl 0698.46050
[13] Haagerup, U., The standard form of von Neumann algebras, Math. Scand., 37, 271-283 (1975) · Zbl 0304.46044
[14] Izumi, M.; Kosaki, H., Kac algebras arising from composition of subfactors: General theory and classification, Mem. Amer. Math. Soc., 158, 750 (2002) · Zbl 1001.46040
[15] Lesieur, F., thesis
[16] Nikshych, D., A duality theorem for quantum groupoids, (New Trends in Hopf Algebra Theory. New Trends in Hopf Algebra Theory, Contemp. Math., vol. 267 (2000)), 237-243 · Zbl 0978.16032
[17] Nikshych, D.; Vainerman, L., Algebraic versions of a finite-dimensional quantum groupoid, (Lect. Notes Pure Appl. Math., vol. 209 (2000)), 189-221 · Zbl 1032.46537
[18] Nikshych, D.; Vainerman, L., A characterization of depth 2 subfactors of \(II_1\) factors, J. Funct. Anal., 171, 2000, 278-307 (2000) · Zbl 1010.46063
[19] Nikshych, D.; Vainerman, L., Finite quantum groupoids and their applications, (New Directions in Hopf Algebras. New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ., vol. 43 (2002), Cambridge University Press), 211-262 · Zbl 1026.17017
[20] Nikshych, D.; Vainerman, L., A Galois correspondence for \(II_1\)-factors and quantum groupoids, J. Funct. Anal., 178, 113-142 (2000) · Zbl 0995.46041
[21] Renault, J., A Groupoid Approach to \(C^\ast \)-Algebras, Lecture Notes in Math., vol. 793 (1980), Springer-Verlag · Zbl 0433.46049
[22] Vallin, J. M., Unitaire pseudo-multiplicatif associé à un groupoïde. Applications à la moyennabilité, J. Operator Theory, 44, 2, 347-368 (2000) · Zbl 0986.22002
[23] Vallin, J. M., Groupoïdes quantiques finis, J. Algebra, 239, 1, 215-261 (2001) · Zbl 1003.46040
[24] Vallin, J. M., Multiplicative partial isometries and finite quantum groupoids, (Proceedings of the Meeting of Theoretical Physicists and Mathematicians. Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg. Proceedings of the Meeting of Theoretical Physicists and Mathematicians. Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, IRMA Lect. Math. Theor. Phys., vol. 2 (2002)), 189-227 · Zbl 1171.47306
[25] Vallin, J. M., Deformation of finite dimensional quantum groupoids
[26] Vaes, S.; Vainerman, L., Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math., 175, 1-101 (2003) · Zbl 1034.46068
[27] Yamanouchi, T., Duality for actions and co-actions of groupoids on von-Neumann algebras, Mem. Amer. Math. Soc., 484 (1993) · Zbl 0822.46070
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