 Circle projective representations lift to unitary representations

The pullback of $U(H)\to PU(H)$ along a continuous circle homomorphism is a compact central extension of the circle. Its lifts commute: a scalar commutator is a continuous alternating bicharacter, which vanishes on a dense cyclic subgroup and hence everywhere. Average a rank-one positive operator over this compact extension. The resulting nonzero compact operator has a finite-dimensional invariant eigenspace, and commuting unitaries have a common eigenvector there. Its continuous character has central weight one; dividing the lifts by that character yields an honest continuous circle representation.