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-Weierstrass theorem. It was first published in 1943.[1] [2]
See also
References
- ^ 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)
- ^ Yoshizawa, Hisaaki. "Unitary representations of locally compact groups. Reproduction of Gelfand-Raikov's theorem." Osaka Mathematical Journal 1.1 (1949): 81-89.