nLab unitary representation



A unitary representation of a locally compact topological group GG on a Hilbert space HH is a representation of GG as an abstract group on HH by unitary linear operators, i.e. the homomorphism of groups ρ:GU(H)EndH\rho : G\to U(H)\subset End H which is continuous with respect to the topology on U(H)U(H) induced by the strong operator topology on EndHEnd H. In other words, ρ:GU(H)\rho: G\to U(H) is a map satisfying ρ(gh)=ρ(g)ρ(h)\rho(g h) = \rho(g)\rho(h), ρ(g) 1=ρ(g 1)=ρ(g) *\rho(g)^{-1} = \rho(g^{-1}) = \rho(g)^* for all g,hGg,h\in G and for all xHx\in H, the function gρ(g)(x)g\mapsto \rho(g)(x) on GG is norm continuous.



  • Jacques Dixmier, Chapter 13 of: C *C^\ast-algebras, North Holland (1977) [ch2:pdf, ch13:pdf]

  • Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press (1995) [pdf, gBooks]

  • Jeffrey Adams, Marc van Leeuwen, Peter Trapa, David A. Vogan Jr, Unitary representations of real reductive groups (arXiv:1212.2192)

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