Type classification of extremal quantized characters

The notion of quantized characters is introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory for compact quantum groups. As in the case of ordinary groups, the representation associated with any extremal quantized character generates von Neumann factor. In the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray-von Neumann-Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_q(N)$.