unitary representation (original) (raw)

Let G be a topological group. A unitary representation of Gis a pair (π,H) where H is a Hilbert space andπ:G→U⁢(H) is a homomorphism such that the mapping of G×H→H that sends (g,v) to π⁢(g)⁢vis continuous. Here U⁢(H) denotes the set of unitary operatorsof H. The group G is said to act unitarily on H or sometimes,G is said to act by unitary representation on H.