TY - JOUR

T1 - Free actions of compact quantum groups on unital C*-algebras

AU - Baum, Paul F.

AU - De Commer, Kenny

AU - Hajac, Piotr M.

N1 - Funding Information:
Acknowledgments. We thank Wojciech Szymański and Makoto Yamashita for helpful and enlightening discussions. We are also very grateful to Jakub Szczepanik for his assistance with LATEX graphics, and to Benjamin Passer and Mariusz Tobolski for their careful proofreading of this paper. This work was partially supported by NCN grant 2011/01/B/ST1/06474. In addition, Paul F. Baum was partially supported by NSF grant DMS 0701184, and Kenny De Commer was partially supported by FWO grant G.0251.15N.

PY - 2017

Y1 - 2017

N2 - Let F be a field, G a finite group, and Map(Γ, F) the Hopf algebra of all set-theoretic maps Γ → F. If E is a finite field extension of F and G is its Galois group, the extension is Galois if and only if the canonical map E ⊗F E → E ⊗ F Map(Γ, F) resulting from viewing E as a Map(Γ, F)-comodule is an isomorphism. Similarly, a finite covering space is regular if and only if the analogous canonical map is an isomorphism. In this paper, we extend this point of view to actions of compact quantum groups on unital C*-algebras. We prove that such an action is free if and only if the canonical map (obtained using the underlying Hopf algebra of the compact quantum group) is an isomorphism. In particular, we are able to express the freeness of a compact Hausdorff topological group action on a compact Hausdorff topological space in algebraic terms. As an application, we show that a field of free actions on unital C*-algebras yields a global free action.

AB - Let F be a field, G a finite group, and Map(Γ, F) the Hopf algebra of all set-theoretic maps Γ → F. If E is a finite field extension of F and G is its Galois group, the extension is Galois if and only if the canonical map E ⊗F E → E ⊗ F Map(Γ, F) resulting from viewing E as a Map(Γ, F)-comodule is an isomorphism. Similarly, a finite covering space is regular if and only if the analogous canonical map is an isomorphism. In this paper, we extend this point of view to actions of compact quantum groups on unital C*-algebras. We prove that such an action is free if and only if the canonical map (obtained using the underlying Hopf algebra of the compact quantum group) is an isomorphism. In particular, we are able to express the freeness of a compact Hausdorff topological group action on a compact Hausdorff topological space in algebraic terms. As an application, we show that a field of free actions on unital C*-algebras yields a global free action.

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M3 - Article

AN - SCOPUS:85034418363

SN - 1431-0635

VL - 22

SP - 825

EP - 849

JO - Documenta Mathematica

JF - Documenta Mathematica

IS - 2017

ER -