nLab unitary representation

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

Examples

• unitary representation of the Poincaré group

Related concepts

• unitarization

• super-unitary representation

References

• Jacques Dixmier, Chapter 13 of: $C^\ast$-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)