Woronowicz shows that the C*-algebras of quantum $SU(2)$ are isomorphic (only as C*-algebras, forgetting the quantum group structure). Are there similar results for quantum $SU(n)$ for $n \geq 3$?
For SU(3), there's . The general case is not fully worked out yet, I guess.
 Nagy, Gabriel; A rigidity property for quantum SU(3) groups. Advances in geometry, 297–336, Progr. Math., 172, Birkhäuser Boston, Boston, MA, 1999.