nLab Gelfand-Raikov theorem

Gel’fand-Raikov theorem (Гельфанд-Райков) The irreducible unitary representations of a locally compact topological group $G$ separates its points. In other words, for any two group elements $g,h\in G$ there exist an irreducible unitary representation $\rho : G\to U(H)$ such that $\rho(g)\neq \rho(h)$.

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press 1995. x+276 pp. gBooks
• И. М. Гельфанд, Д. А. Райков, Неприводимые унитарные представления локально бикомпактных групп, Матем. сб., 13(55):2-3 (1943), 301–316, pdf (I. Gelfand, D. Raikov, “Irreducible unitary representations of locally bicompact groups”, Rec. Math. N.S., 13(55):2-3 (1943), 301–316)