Gelfand–Raikov theorem

The Gel’fand-Raikov (Гельфанд-Райков) theorem states that the points of locally compact topological group G are separated by its irreducible unitary representations. In other words, of for any two group elements g,h∈G there exist an irreducible unitary representation ρ:G→U(H) such that ρ(g)≠ρ(h). It follows that on every compact subset of the group, the continuous functions defined by the matrix elements <e_i, ρ(g)e_j> with e_i orthonormal basis vectors in H, are dense in the space of continuous functions by the Stone-von Neumann theorem. It was first published in 1943.