# A Construction of Finite Index C*-algebra Inclusions from Free Actions of Compact Quantum Groups

### Kenny De Commer

Université de Cergy-Pontoise, France### Makoto Yamashita

Ochanomizu University, Tokyo, Japan

## Abstract

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.

## Cite this article

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. 709–735

DOI 10.4171/PRIMS/117