Free Actions of Compact Quantum Groups on Unital C*-Algebras

Let $F$ be a field, $\\Gamma$ a finite group, and ${Map}(\Gamma,F)$
the Hopf algebra of all set-theoretic maps $\Gamma\rightarrow F$. If $E$
is a finite field extension of $F$ and $\Gamma$ is its Galois group, the
extension is Galois if and only if the canonical map $E\otimes_{F}E\rightarrow
E\otimes_{F}{Map}(\Gamma,F)$ resulting from viewing $E$ as a ${Map}(\Gamma,F)$-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.

2010 Mathematics Subject Classification:

Keywords and Phrases:

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