We compare algebraic objects related to a compact quantum group action on a unital $C^∗$-algebra in the sense of Podleś and Baum et al. and show that they differ by the kernel of the morphism describing the action. Then we address ways to remove the kernel without changing the Podleś algebraic core. A minimal such procedure is described. We end the paper with a natural example of an action of a reduced compact quantum group with non-trivial kernel.
Piotr M. Sołtan. "On actions of compact quantum groups." Illinois J. Math. 55 (3) 953 - 962, Fall 2011. https://doi.org/10.1215/ijm/1369841792