Residually finite quantum group algebras

We show that provided n≠3, the involutive Hopf ⁎-algebra Au(n) coacting universally on an n-dimensional Hilbert space has enough finite-dimensional representations in the sense that every non-zero element acts non-trivially in some finite-dimensional ⁎-representation. This implies that the discrete quantum group with group algebra Au(n) is maximal almost periodic (i.e. it embeds in its quantum Bohr compactification), answering a question posed by P. Sołtan in [21].We also prove analogous results for the involutive Hopf ⁎-algebra Bu(n) coacting universally on an n-dimensional Hilbert space equipped with a non-degenerate bilinear form.