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Free actions of compact quantum groups on unital $$C^\ast$$-algebras. (English) Zbl 1386.46055
The theorem of this paper is stated as follows. Let $$A$$ be a unital $$C^*$$-algebra with an action of a compact quantum group $$H_q$$ with co-multiplication $$\Delta: H_q\rightarrow H_q \otimes H_q$$, with the coaction $$\delta$$ as an injective unital $$\ast$$-homomorphism from $$A$$ to $$A \otimes H_q$$. Then there are equivalences among: (1) the action of $$H_q$$ on $$A$$ is free; (2) the action of $$H_q$$ on $$A$$ satisfies the Peter-Weyl-Galois (PWG) condition; (3) the action of $$H_q$$ on $$A$$ is strongly monoidal.
Also provided by the theorem above is the following characterization. Let $$G$$ be a compact Hausdorff group acting continuously on a compact Hausdorff space $$X$$. Then the action of $$G$$ on $$X$$ is free if and only if the canonical map, defined below, is an isomorphism.
Recall several definitions concerning the first statement as follows. By definition, the coaction $$\delta$$ satisfies both the co-associativity related to $$\Delta$$ and the co-unitality (or density) in $$A\otimes H_q$$ related to $$A$$ and $$H_q$$. By definition, the coaction is free if the density in $$A\otimes H_q$$ related to only $$A$$ holds.
Moreover, for a compact quantum group $$H_q$$, there is its dense Hopf $$\ast$$-subalgebra $$H_f$$ spanned by the matrix coefficients of its irreducible unitary representations. Then the Peter-Weyl subalgebra $$PW_{H_q}(A)$$ of $$A$$ is defined as the inverse image of $$\delta$$ in $$A\otimes H_f$$. By definition, the coaction satisfies the PWG condition if the canonical map from $$PW_{H_q}(A)\otimes_B PW_{H_q}(A)$$ to $$PW_{H_q}(A) \otimes H_f$$ involving $$\delta$$ is bijective, where $$B$$ is the unital (fixed point) $$C^*$$-subalgebra of $$A$$ of coaction invariants $$a\in A$$ so that $$\delta(a)= a\otimes 1$$.
By definition, the coaction $$\delta$$ is strongly monoidal if the map extended from the canonical map above by involving co-tensor products with any left $$H_f$$-comodules $$V$$ and $$W$$ as well as $$V \otimes W$$, respectively on the left and right sides, is bijective.

##### MSC:
 46L55 Noncommutative dynamical systems 46L05 General theory of $$C^*$$-algebras 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 16T05 Hopf algebras and their applications 16T20 Ring-theoretic aspects of quantum groups 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
##### Keywords:
$$C^\ast$$-algebra; quantum group; free action; Hopf algebra
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##### References:
 [1] Artin M., Grothendieck A. and Verdier J.-L.,Th\'eorie des Topos et Cohomologie \'etale des Sch\'emas, Springer Lecture Notes in Math. 269, 1972. [2] Atiyah M. F. and Hirzebruch F., Riemann-Roch theorems for differentiable manifolds. Bull. Amer. Math. Soc. 65 (1959) 276-281. · Zbl 0142.40901 [3] Atiyah M. F. and Hirzebruch F., Vector bundles and homogeneous spaces. Differential geometry. Proceedings of the Symposium in Pure Mathematics, Vol. III, American Mathematical Society, Providence, R.I., 7-38, 1961. · Zbl 0108.17705 [4] Baum P. F. and Hajac P. M., Local proof of algebraic characterization of free actions. SIGMA 10 (2014), 060, 7 pages. · Zbl 1295.22010 [5] Beggs E. J. and Brzezi\'nski T., An explicit formula for a strong connection. Applied Categorical structures 16 (2008) 57-63. · Zbl 1195.16030 [6] Brzezi\'nski T. and Hajac P. M., The Chern-Galois character. C. R. Acad. Sci. Paris, Ser. I 338 (2004) 113-116. · Zbl 1061.16037 [7] Brzezi\'nski T. and Hajac P. M., Galois-type extensions and equivariant projectivity. In: Quantum Symmetry in Noncommutative Geometry, P. M. Hajac (ed.), Eur. Math. Soc. Publ. House, to appear. [8] Connes A., Noncommutative Geometry, Academic Press, San Diego, CA, 1994, 661 pages, ISBN 0-12-185860-X. [9] D֒abrowski L., Hadfield T. and Hajac P. M., Equivariant join and fusion of noncommutative algebras. SIGMA 11 (2015), 082, 7 pages. DHS99] D\k abrowski L., Hajac P. M. and Siniscalco P., Explicit Hopf-Galois description of SLe2iπ3(2)-induced Frobenius homomorphisms. Enlarged Proceedings of the ISI GUCCIA Workshop on quantum groups, non commutative geometry and fundamental physical interactions, D. Kastler, M. Rosso, T. Schucker (eds.), Commack, New York, Nova Science Pub., Inc., 279-298, 1999. [10] De Commer K. and Yamashita M., A construction of finite index $$C^*$$-algebra inclusions from free actions of compact quantum groups. Publ. Res. Inst. Math. Sci. 49 (2013) 709-735. · Zbl 1338.46077 [11] Demazure M. and Gabriel P., Groupes Alg ́ebriques I, North Holland, Amsterdam, 1970. [12] Dupr ́e M. J. and Gillette R. M., Banach bundles, Banach modules and automorphisms of $$C^*$$-algebras. Research Notes in Math. 92,Pitman Advanced Pub. Program, 1983. [13] Ellwood D. A., A new characterisation of principal actions. J. Funct. Anal. 173 (2000) 49-60. · Zbl 0960.46049 [14] Hajac P. M., Strong connections on quantum principal bundles. Comm. Math. Phys. 182 (1996) 579-617. · Zbl 0873.58007 [15] Hajac P. M., Kr\"ahmer U., Matthes R. and Zieli\'nski B., Piecewise principal comodule algebras. J. Noncommut. Geom. 5 (2011) 591-614. · Zbl 1258.16036 [16] Hajac P. M., Matthes R. and Szyma\'nski W., Chern numbers for two families of noncommutative Hopf fibrations. C. R. Acad. Sci. Paris, Ser. I 336 (2003) 925-930. · Zbl 1029.46112 [17] Kasparov G. G., Equivariant KK-theory and the Novikov conjecture. Inv. Math. 91 (1988) 147-201. · Zbl 0647.46053 [18] Lance E. C., Hilbert C∗-modules. A toolkit for operator algebraists. London Math. Soc. Lecture Note Series 210, Cambridge Univ. Press, Cambridge, 1995. [19] Maes A. and Van Daele A., Notes on compact quantum groups. Nieuw Arch. Wisk. 16 (1998) 73-112. · Zbl 0962.46054 [20] Podle ́s P., Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups. Comm. Math. Phys. 170 (1995) 1-20. [21] Rieffel M. A., Continuous fields of $$C^*$$-algebras coming from group cocycles and actions. Math. Ann. 283 (1989) 631-643. · Zbl 0646.46063 [22] Schauenburg P., Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules. Applied Categorical Structures 6 (1998) 193-222. · Zbl 0908.16033 [23] Schauenburg P., Hopf-Galois and Bi-Galois extensions. Galois theory, Hopf algebras, and semiabelian categories, Fields Inst. Comm. 43, AMS 2004, 469-515. · Zbl 1091.16023 [24] Schauenburg P. and Schneider H.-J., On generalized Hopf Galois extensions. J. Pure Appl. Algebra 202 (2005) 168-194. · Zbl 1081.16045 [25] Schneider H.-J., Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. of Math. 72 (1990) 167-195. · Zbl 0731.16027 [26] So ltan P. M., On actions of compact quantum groups. Illinois Journal of Mathematics 55 (2011) 953-962. [27] Swan R. G., Vector bundles and projective modules. Trans. Amer. Math. Soc. 105 (1962) 264-277. · Zbl 0109.41601 [28] Ulbrich K.-H., Fibre functors of finite-dimensional comodules. Manuscripta Math. 65 (1989) 39-46. · Zbl 0674.16006 [29] Woronowicz S. L., Compact matrix pseudogroups. Comm. Math. Phys. 111 (1987) 613-665. · Zbl 0627.58034 [30] Woronowicz S. L., Compact quantum groups. Sym\'etries quantiques (Les Houches, 1995) , 845-884, North-Holland, Amsterdam, 1998. · Zbl 0997.46045
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