Compact quantum group actions on compact quantum spaces
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In this dissertation, we study compact quantum group actions on compact quantum spaces, i.e., coactions on unital C *-algebras. Through analyzing invariant subsets and invariant states, we show that as long as all invariant states are tricial or there exists a faithful tracial invariant state, the compact quantum group is a Kac algebra. This result generalizes lots of results in the literature. Then we investigate compact quantum group actions on compact Hausdorff spaces. We construct faithful actions of quantum permutation groups on connected compact metrizable spaces, which provide the first examples of genuine compact quantum group actions on connected compact metrizable spaces and disproves a conjecture of Goswami. Next we formulate the concept of compact quantum group orbits which is a good generalization of classical orbits. Later, we develop orbits as a tool to study classical compact quantum homogeneous spaces, and show that the unique invariant measure of a compact quantum homogeneous space with infinitely many points is non-atomic. It turns out that countable compact metrizable spaces with infinitely many points are not quantum homogeneous spaces.