## Banach Journal of Mathematical Analysis

### Partial actions of $C^{*}$-quantum groups

#### Abstract

Partial actions of groups on $C^{*}$-algebras and the closely related actions and coactions of Hopf algebras have received much attention in recent decades. They arise naturally as restrictions of their global counterparts to noninvariant subalgebras, and the ambient enveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of $C^{*}$-bialgebras, focusing on $C^{*}$-quantum groups, and we prove the existence of an enveloping global coaction under mild technical assumptions. We also show that partial coactions of the function algebra of a discrete group correspond to partial actions on direct summands of a $C^{*}$-algebra, and we relate partial coactions of a compact or its dual discrete $C^{*}$-quantum group to partial coactions or partial actions of the dense Hopf subalgebra. As a fundamental example, we associate to every discrete $C^{*}$-quantum group a quantum Bernoulli shift.

#### Article information

Source
Banach J. Math. Anal., Volume 12, Number 4 (2018), 843-872.

Dates
Accepted: 1 September 2017
First available in Project Euclid: 30 January 2018

https://projecteuclid.org/euclid.bjma/1517281422

Digital Object Identifier
doi:10.1215/17358787-2017-0056

Mathematical Reviews number (MathSciNet)
MR3858752

Zentralblatt MATH identifier
06946294

#### Citation

Kraken, Franziska; Quast, Paula; Timmermann, Thomas. Partial actions of $C^{*}$ -quantum groups. Banach J. Math. Anal. 12 (2018), no. 4, 843--872. doi:10.1215/17358787-2017-0056. https://projecteuclid.org/euclid.bjma/1517281422

#### References

• [1] F. Abadie, Enveloping actions and Takai duality for partial actions, J. Funct. Anal. 197 (2003), no. 1, 14–67.
• [2] F. Abadie, Sobre ações parciais, fibrados de Fell e grupóides, preprint, Ph.D. dissertation, University of Sáo Paulo, Sáo Paulo, Brazil, 1999.
• [3] E. R. Alvares, M. M. S. Alves, and E. Batista, Partial Hopf module categories, J. Pure Appl. Algebra 217 (2013), no. 8, 1517–1534.
• [4] M. M. S. Alves and E. Batista, Enveloping actions for partial Hopf actions, Comm. Algebra 38 (2010), no. 8, 2872–2902.
• [5] M. M. S. Alves and E. Batista, “Globalization theorems for partial Hopf (co)actions, and some of their applications” in Groups, Algebras and Applications, Contemp. Math. 537, Amer. Math. Soc., Providence, 2011, 13–30.
• [6] M. M. S. Alves, E. Batista, M. Dokuchaev, and A. Paques, Globalization of twisted partial Hopf actions, J. Aust. Math. Soc. 101 (2016), no. 1, 1–28.
• [7] M. M. S. Alves, E. Batista, and J. Vercruysse, Partial representations of Hopf algebras, J. Algebra 426 (2015), 137–187.
• [8] S. Baaj and G. Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de $C^{*}$-algèbres, Ann. Sci. Éc. Norm. Supér. (4) 26 (1993), no. 4, 425–488.
• [9] S. Baaj, G. Skandalis, and S. Vaes, Non-semi-regular quantum groups coming from number theory, Comm. Math. Phys. 235 (2003), no. 1, 139–167.
• [10] E. Bédos and L. Tuset, Amenability and co-amenability for locally compact quantum groups, Int. J. Math. 14 (2003), no. 8, 865–884.
• [11] B. Blackadar, Operator Algebras: Theory of $C^{*}$-Algebras and von Neumann Algebras, Encyclopaedia Math. Sci. 122, Springer, Berlin, 2006.
• [12] S. Caenepeel and K. Janssen, Partial (co)actions of Hopf algebras and partial Hopf–Galois theory, Comm. Algebra 36 (2008), no. 8, 2923–2946.
• [13] F. Castro, A. Paques, G. Quadros, and A. Santána, Partial actions of weak Hopf algebras: Smash product, globalization and Morita theory, J. Pure Appl. Algebra 219 (2015), no. 12, 5511–5538.
• [14] R. Exel, Circle actions on $C^{*}$-algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence, J. Funct. Anal. 122 (1994), no. 2, 361–401.
• [15] R. Exel, Twisted partial actions: A classification of regular $C^{*}$-algebraic bundles, Proc. Lond. Math. Soc. (3) 74 (1997), no. 2, 417–443.
• [16] R. Exel, Partial dynamical systems, Fell bundles and applications, Math. Surveys Monogr. 224, Amer. Math. Soc., Providence, 2017.
• [17] U. Franz and A. Skalski, On idempotent states on quantum groups, J. Algebra 322 (2009), no. 5, 1774–1802.
• [18] J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), no. 6, 837–934.
• [19] E. C. Lance, Hilbert $C^{*}$-Modules: A Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Ser. 210, Cambridge Univ. Press, Cambridge, 1995.
• [20] M. B. Landstad and A. Van Daele, Groups with compact open subgroups and multiplier Hopf $*$-algebras, Expo. Math. 26 (2008), no. 3, 197–217.
• [21] K. McClanahan, $K$-theory for partial crossed products by discrete groups, J. Funct. Anal. 130 (1995), no. 1, 77–117.
• [22] R. Meyer, S. Roy, and S. L. Woronowicz, Quantum group-twisted tensor products of $C^{*}$-algebras, Int. J. Math. 25 (2014), no. 2, art. ID rnm1450019.
• [23] S. Neshveyev and L. Tuset, Compact Quantum Groups and Their Representation Categories, Cours Spéc 20, Soc. Math. France, Paris, 2013.
• [24] S. Roy, $C^{*}$-quantum groups with projection, preprint, Ph.D. dissertation, University of Göttingen, Göttingen, Germany, 2013.
• [25] S. Roy and T. Timmermann, The maximal quantum group-twisted tensor product of $C^{*}$-algebras, to appear in J. Noncommut. Geom., preprint, arXiv:1511.06100v3 [math.OA].
• [26] P. M. Sołtan, Quantum families of maps and quantum semigroups on finite quantum spaces, J. Geom. Phys. 59 (2009), no. 3, 354–368.
• [27] P. M. Sołtan and S. L. Woronowicz, A remark on manageable multiplicative unitaries, Lett. Math. Phys. 57 (2001), no. 3, 239–252.
• [28] P. M. Sołtan and S. L. Woronowicz, From multiplicative unitaries to quantum groups, II, J. Funct. Anal. 252 (2007), no. 1, 42–67.
• [29] M. Takesaki and N. Tatsuuma, Duality and subgroups, Ann. of Math. (2) 93 (1971), 344–364.
• [30] O. Uuye and J. Zacharias, The Fubini product and its applications, preprint, arXiv:1603.04527v1 [math.OA].
• [31] S. Wassermann, The slice map problem for $C^{*}$-algebras, Proc. Lond. Math. Soc. (3) 32 (1976), no. 3, 537–559.
• [32] S. L. Woronowicz, From multiplicative unitaries to quantum groups, Int. J. Math. 7 (1996), no. 1, 127–149.
• [33] S. L. Woronowicz, “Compact quantum groups” in Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 845–884.