Spectra for compact quantum group coactions and crossed products

We present definitions of both Connes spectrum and strong Connes spectrum for actions of compact quantum groups on C*-algebras and obtain necessary and sufficient conditions for a crossed product to be a prime or a simple C*-algebra. Our results extend to the case of compact quantum actions the results in [8] which in turn, generalize results by Connes, Olesen and Pedersen and Kishimoto for abelian group actions. We prove in addition that the Connes spectra are closed under tensor products. These results are new for compact nonabelian groups as well.