Definition 0.1   A compact quantum group, $Q_{CG}$ is defined as a particular case of a locally compact quantum group $Q_{LCG}$ , that is, a quadruple $(A, \Delta, \mu, \nu)$ , where $A$ is either a $C^*$ - or a $W^*$ - algebra equipped with a co-associative comultiplication $\Delta: A \to A \otimes A$ , and two faithful semi-finite normal weights, $\mu$ and $\nu$ -right and -left Haar measures, and also when the object space $\mathbf{O}$ of the latter $Q_{LCG}$ is replaced by a compact topological space $Q^T_{CG}$ , instead of being a locally compact topological space like $Q_{LCG}$ .