Hilbert space representations of cross product algebras

Hilbert space representations of cross product ∗-algebras of the Hopf ∗-algebras Uq(gl2) with the coordinate algebras Ô(Cq2) and O(Rq3) of quantum vector spaces, and of Uq(su2) with the coordinate algebras O(SUq(2)) and O(Sq2) of the corresponding quantum spheres, are investigated and classified. Invariant states on the coordinate ∗-algebras are described by two variants of the quantum trace