Gelfand–Raikov theorem explained
The Gel'fand–Raikov (Гельфанд–Райков) theorem is a theorem in the mathematics of locally compact topological groups. It states that a locally compact group is completely determined by its (possibly infinite dimensional) unitary representations. The theorem was first published in 1943.[1] [2]
A unitary representation
of a locally compact group
on a
Hilbert space
defines for each pair of vectors
a continuous function on
, the
matrix coefficient, by
g\mapsto\langleh,\rho(g)k\rangle
. The set of all matrix coefficientsts for all unitary representations is closed under scalar multiplication (because we can replace
), addition (because of
direct sum representations), multiplication (because of
tensor representations) and complex conjugation (because of the
complex conjugate representations).
The Gel'fand–Raikov theorem now states that the points of
are separated by its irreducible unitary representations, i.e. for any two group elements
there exist a Hilbert space
and an irreducible unitary representation
such that
. The matrix elements thus separate points, and it then follows from the
Stone–Weierstrass theorem that on every compact subset of the group, the matrix elements are dense in the space of continuous functions, which determine the group completely.
See also
Notes and References
- http://www.mathnet.ru/php/getFT.phtml?jrnid=sm&paperid=6181&volume=55&year=1943&issue=2&fpage=301&what=fullt И. М. Гельфанд, Д. А. Райков, Неприводимые унитарные представления локально бикомпактных групп, Матем. сб., 13(55):2–3 (1943), 301–316
- http://projecteuclid.org/download/pdf_1/euclid.ojm/1200685627 Yoshizawa, Hisaaki. "Unitary representations of locally compact groups. Reproduction of Gelfand–Raikov's theorem." Osaka Mathematical Journal 1.1 (1949): 81–89