  We study actions of compact quantum groups on type I factors, which may be
interpreted as projective representations of compact quantum groups. We
generalize to this setting some of Woronowicz' results concerning Peter-Weyl
theory for compact quantum groups. The main new phenomenon is that for general
compact quantum groups (more precisely, those which are not of Kac type), not
all irreducible projective representations have to be finite-dimensional. As
applications, we consider the theory of projective representations for the
compact quantum groups associated to group von Neumann algebras of discrete
groups, and consider a certain non-trivial projective representation for
quantum SU(2).
