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

Spectra for compact quantum group coactions and crossed products