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Abstract

We study certain principal actions on noncommutative $C^*$-algebras. Our main examples are the $\mathbb Z_p$- and $\mathbb T$-actions on the odd-dimensional quantum spheres, yielding as fixed-point algebras quantum lens spaces and quantum complex projective spaces, respectively. The key tool in our analysis is the relation of the ambient $C^*$-algebras with the Cuntz-Krieger algebras of directed graphs. A general result about the principality of the gauge action on graph algebras is given.