Given an action of a compact quantum group on a unital C-algebra, one can consider the associated Wassermann-type C-algebra inclusions. One hereby amplifies the original action with the adjoint action associated with a finite dimensional unitary representation, and considers the induced inclusion of fixed point algebras. We show that this inclusion is a finite index inclusion of C-algebras when the quantum group acts freely. Along the way, two natural definitions of freeness for a compact quantum group action, due respectively to D. Ellwood and M. Rieffel, are shown to be equivalent.
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Kenny De Commer, Makoto Yamashita, A Construction of Finite Index C*-algebra Inclusions from Free Actions of Compact Quantum Groups. Publ. Res. Inst. Math. Sci. 49 (2013), no. 4, pp. 709–735DOI 10.4171/PRIMS/117