Gelfand–Raikov theorem

The Gel'fand–Raikov (Гельфанд–Райков) theorem is a theorem in the theory of locally compact topological groups. It states that a locally compact group is completely determined by its (possibly infinite dimensional) unitary representations. The theorem was first published in 1943.[1] [2]

A unitary representation of a locally compact group G defines a set of continuous functions on G by <e_i, ρ(g)e_j> where {e_i} is some basis of orthonormal vectors in H (the matrix coefficients). The set of matrix elements for all unitary representations is invariant under complex conjugation because of the existence of the complex conjugate representation on ${\displaystyle {\bar {H}}}$.

The Gel'fand–Raikov theorem now states that the points of G are separated by its irreducible unitary representations, i.e. for any two group elements g,h ∈ G there exist a Hilbert space H and an irreducible unitary representation ρ : G → U(H) such that ρ(g) ≠ ρ(h). The matrix elements thus separate points and it then follows from the Stone–Weierstrass theorem that on every compact subset of the group, the matrix elements are dense in the space of continuous functions, which determine the group completely.