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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)
Full Text: DOI
[1] Aguiar, M.; Andruskiewitsch, N., Representations of matched pairs of groupoids and applications to weak Hopf algebras, () · 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 \(\operatorname{C}^\ast\)-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 \(\operatorname{C}^\ast\)-Hopf algebras: the coassociative symmetry of non-integral dimensions, (), 9-19 · Zbl 0894.16018
[7] Böhm, G.; Szlachányi, K.; Nill, F., Weak Hopf algebras I. integral theory and \(\operatorname{C}^\ast\)-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 · 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)
[15] Lesieur, F., thesis
[16] Nikshych, D., A duality theorem for quantum groupoids, (), 237-243 · Zbl 0978.16032
[17] Nikshych, D.; Vainerman, L., Algebraic versions of a finite-dimensional quantum groupoid, (), 189-221 · Zbl 1032.46537
[18] Nikshych, D.; Vainerman, L., A characterization of depth 2 subfactors of \(\mathit{II}_1\) factors, J. funct. anal., 171, 2000, 278-307, (2000) · Zbl 1010.46063
[19] Nikshych, D.; Vainerman, L., Finite quantum groupoids and their applications, (), 211-262 · Zbl 1026.17017
[20] Nikshych, D.; Vainerman, L., A Galois correspondence for \(\mathit{II}_1\)-factors and quantum groupoids, J. funct. anal., 178, 113-142, (2000) · Zbl 0995.46041
[21] Renault, J., A groupoid approach to \(\operatorname{C}^\ast\)-algebras, Lecture notes in math., vol. 793, (1980), Springer-Verlag
[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)
[24] Vallin, J.M., Multiplicative partial isometries and finite quantum groupoids, (), 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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