Knowlpedia

Weak Continuity of a Representation

Continuity of matrix coefficients (π(g)x,y) in the group parameter g

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

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

Remarks

Key property (paper use):

  • Theorem 3.1 proves weak continuity of U\mathfrak U; Theorem 4.2 lifts this to Y‾\overline{Y}.
Examples
  • Any strongly continuous unitary representation is weakly continuous.