Weak Continuity of a Representation

A unitary representation $\pi:G\to U(\mathcal H)$ is weakly continuous if $g\mapsto (\pi(g)x,y)$ is continuous for all $x,y\in\mathcal H$.

For a projective representation $\overline{\pi}$, Shale uses: $g\mapsto |(\overline{\pi}(g)x,y)|$ continuous.

Key property (paper use):

• Theorem 3.1 proves weak continuity of $\mathfrak U$; Theorem 4.2 lifts this to $\overline{Y}$.

Example: Any strongly continuous unitary representation is weakly continuous.