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 infinity dimensional) unitary representations.

More precisely, the points of a locally compact topological group G are separated by its irreducible unitary representations. In other words, 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 then follows from the Stone–Weierstrass theorem that on every compact subset of the group, the continuous functions defined by <ei, ρ(g)ej> with e_i orthonormal basis vectors in H (the matrix coefficients), are dense in the space of continuous functions. The theorem was first published in 1943.^{[1] [2]}

See also

• Gelfand–Naimark theorem
• Representation theory

References

1. И. М. Гельфанд, Д. А. Райков, Неприводимые унитарные представления локально бикомпактных групп, Матем. сб., 13(55):2–3 (1943), 301–316, (I. Gelfand, D. Raikov, "Irreducible unitary representations of locally bicompact groups", Rec. Math. N.S., 13(55):2–3 (1943), 301–316)
2. Yoshizawa, Hisaaki. "Unitary representations of locally compact groups. Reproduction of Gelfand–Raikov's theorem." Osaka Mathematical Journal 1.1 (1949): 81–89.