  Hilbert space representations of cross product *-algebras of the Hopf
*-algebras U_q(gl_2) with the coordinate algebras O(C^2_q) and O(R^3_q) of
quantum vector spaces and U_q(su_2) with coordinate algebras O(SU_q(2)) and
O(S^2_q) of corresponding quantum spheres are investigated and classified.
Invariant states on the coordinate *-algebras are described by two variants of
the quantum trace.
