Fixed points of compact quantum groups actions on Cuntz algebras

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").

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