nLab Gelfand-Raikov theorem

Idea

The Gel’fand-Raikov theorem (Гельфанд-Райков) says that the irreducible unitary representations of a locally compact topological group $G$ separate its points.

In other words, for any two group elements $g, h\in G$ there exist an irreducible unitary representation $\rho \colon G \to U(H)$ such that $\rho(g)\neq \rho(h)$.

Related concepts

• state on a star-algebra

References

The characterization of states on group algebras and what came to be known as the Gelfand-Raikov theorem:

• И. М. Гельфанд, Д. А. Райков, Неприводимые унитарные представления локально бикомпактных групп, Матем. сб., 13(55):2–3 (1943) 301–316 [mathnet pdf]

  
  

  Israel Gelfand, Dmitri Raikov, Irreducible unitary representations of locally bicompact groups, Recueil Mathématique. N.S., 13(55) 2–3 (1943) 301–316 [mathnet:eng/sm6181]

See also

• Wikipedia, Gelfand-Raikov theorem

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press (1995) [pdf, gBooks]