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.

Remarks

Key property (paper use):

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

Examples