 Unitary representation

A unitary representation of a group $G$ on a complex Hilbert space $H$ is a group homomorphism $U:G\to \mathcal{U}(H)$ into its unitary operators. Thus $U(gh)=U(g)U(h)$ and $⟨U(g)v,U(g)w⟩=⟨v,w⟩$. In finite dimension, this is the condition $U(g{)}^{†}U(g)=I$ in an orthonormal basis.

For a topological group, one normally also requires strong continuity: $g\mapsto U(g)v$ is continuous for every vector $v$. This gives a continuous group action on $H$ and permits the integrated unitary representation. For example, $(U(q)f)(t)=f(t+q)$ is a strongly continuous unitary representation of the additive real numbers on $L^2(\mathbb{R})$; preservation of the inner product follows from translation invariance of the Lebesgue measure.