Projective unitary group ($PU(H)=U(H)/\mathbb{T}I$)

The quotient identifies unitary operators differing by a scalar phase. A continuous projective unitary representation takes values in this quotient, with the quotient of the strong operator topology. A local continuous section fixes the phase by requiring a chosen nonzero matrix coefficient to be positive real.

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.