  Given an action of a Compact Quantum Group (CQG) on a finite dimensional
Hilbert space, we can construct an action on the associated Cuntz algebra. We
study the fixed point algebra of this action, using Kirchberg classification
results. Under certain conditions, we prove that the fixed point algebra is
purely infinite and nuclear. We further identify it as a Pimsner algebra,
compute its $K$-theory and prove a "stability property": the fixed points only
depend on the CQG via its fusion rules. We apply the theory to $SU_q(N)$ and
illustrate by explicit computations for $SU_q(2)$ and $SU_q(3)$. This
construction provides examples of free actions of CQG (or "principal
noncommutative bundles").
