Polynomial growth of discrete quantum groups, topological dimension of the dual and $^*$-regularity of the Fourier algebra

Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand-Kirillov dimension of an algebra. Furthermore, we show that polynomial growth for a compact quantum group $G$ of Kac type implies $^*$--regularity of the Fourier algebra $A(G)$, that is every closed ideal of $C(G)$ has a dense intersection with $A(G)$. In particular, $A(G)$ has a unique $C^*$--norm.