Hilbert Space Representations of Cross Product Algebras

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.