2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61515Hilbert 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.LaTeX2e, 44 pages; appears in Journal of Functional AnalysisQuantum AlgebraOperator Algebras17B37, 81R50, 47L55text