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. A unitary representation formula_1 of a locally compact group formula_2 on a Hilbert space formula_3 defines for each pair of vectors formula_4 a continuous function on formula_2, the matrix coefficient, by The set of all matrix coefficientsts for all unitary representations is closed under scalar multiplication (because we can replace formula_7), addition (because of direct sum  representations), multiplication (because of tensor representations) and complex conjugation (because of the complex conjugate representations). The Gel'fand–Raikov theorem now states that the points of formula_2 are separated by its irreducible unitary representations, i.e. for any two group elements formula_9 there exist a Hilbert space formula_10 and an irreducible unitary representation formula_1 such that formula_12. 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.