Projective unitary representation in the Shale paper

A projective unitary representation assigns each $g\in G$ a unitary ray $\overline{U}(g)=\{\alpha U(g):|\alpha|=1\}$ , with multiplication holding up to a phase (a cocycle).

Remarks

Key properties (paper use):

Shale's implementers $Y(T)$ are unique only up to phase, so $\overline{Y}$ is projective.
In finite dimensions, $\overline{Y}$ lifts to a genuine double-valued unitary representation (§5).

Examples

Spin representations are projective representations of rotation groups.