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 $\rho :G\to U(H)\subset \mathrm{End}H$ which is continuous with respect to the topology on $U(H)$ induced by the strong operator topology on $\mathrm{End}H$. In other words, $\rho :G\to U(H)$ is a map satisfying $\rho (\mathrm{gh})=\rho (g)\rho (h)$, $\rho (g{)}^{-1}=\rho (g^{-1})=\rho (g{)}^{*}$ 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

References

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995

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