Journal of Lie Theory

We consider three quantum algebras: the $q$-oscillator algebra, the Podle\'s sphere and the $q$-deformed enveloping algebra of $su(2)$. To each of these $*$-algebras we associate a certain partial dynamical system and perform the ``Mackey analysis'' of $*$-representations developed by Yu. Savchuk and K. Schm\"udgen [``Unbounded induced representations of $*$-algebras'', Algebr. Represent. Theory, DOI: 10.1007/s10468-011-9310-6]. As a result we get the description of ``standard'' irreducible $*$-representations. Further, for each of these examples we show the existence of a ``$C^*$-envelope'' which is canonically isomorphic to the covariance $C^*$-algebra of the partial dynamical system. Finally, for the $q$-oscillator algebra and the $q$-deformed ${\cal U}(su(2))$ we show the existence of ``bad'' representations.