Polynomial growth for compact quantum groups, topological dimension and *regularity of the Fourier algebra

Banica and Vergnioux have shown that the dual discrete quantum group of acompact 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, byconnecting 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.