Hilbert space representations of cross product algebras II

In this paper, we study and classify Hilbert space representations of cross product *-algebras of the quantized enveloping algebra $U_q(e_2)$ with the coordinate algebras $O(E_q(2))$ of the quantum motion group and $O({\text{C}}_q)$ of the complex plane, and of the quantized enveloping algebra $U_q(s{u}_{1,1})$ with the coordinate algebras $O(S{U}_q(1,1))$ of the quantum group $S{U}_q(1,1)$ and $O(U_q)$ of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitely described.