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Let F be a field, G a finite group, and Map(G,F) the Hopf algebra of

all set-theoretic maps G -> 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 resulting from

viewing E as a Map(G,F)-comodule is an isomorphism. Similarly, a

finite covering space is regular if and only if the analogous

canonical map is an isomorphism. The main result to be presented in

this talk is an extension of this point of view to arbitrary actions

of compact quantum groups on unital C*-algebras. I will explain that

such an action is free (in the sense of Ellwood) 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.

Also, we can apply the main result to noncommutative join

constructions and coactions of discrete groups on unital C*-algebras.

(Joint work with Paul F. Baum and Kenny De Commer.)

Free actions of compact quantum group on unital C*-algebras