Gelfand–Raikov theorem
The Gel’fand-Raikov (Гельфанд-Райков) theorem states that the points of a 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 coefficients <ei, ρ(g)ej> with ei 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[1].
References
See Also
- ^ 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)