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

Abstract

Given an action of a compact quantum group on a unital C${}^{\ast }$-algebra, one can consider the associated Wassermann-type C${}^{\ast }$-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${}^{\ast }$-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.