About: Gelfand–Raikov theorem

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The Gel'fand–Raikov (Гельфанд–Райков) theorem is a theorem in the theory 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. 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 .

Attributes	Values
rdfs:label	Gelfand–Raikov theorem (en)
rdfs:comment	The Gel'fand–Raikov (Гельфанд–Райков) theorem is a theorem in the theory 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. 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 . (en)
dct:subject	Representation theory of groups
Wikipage page ID	43386258 (xsd:integer)
Wikipage revision ID	1080941019 (xsd:integer)
Link from a Wikipage to another Wikipage	Representation theory Complex conjugate representation Gelfand–Naimark theorem Locally compact topological group Stone–Weierstrass theorem Matrix coefficient Direct sum Hilbert space Representation theory of groups Unitary representation Tensor representation
sameAs	Gelfand–Raikov theorem Gelfand–Raikov theorem Gelfand–Raikov theorem
dbp:wikiPageUsesTemplate	dbt:Reflist dbt:Context dbt:Topology-stub
has abstract	The Gel'fand–Raikov (Гельфанд–Райков) theorem is a theorem in the theory 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. 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 . 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. (en)
gold:hypernym	Theorem
prov:wasDerivedFrom	wikipedia-en:Gelfand–Raikov_theorem?oldid=1080941019&ns=0
page length (characters) of wiki page	2846 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Gelfand–Raikov_theorem
is Link from a Wikipage to another Wikipage of	Gelfand–Naimark theorem Gelfand-Raikov theorem Topological group Gelfand Raikov theorem
is Wikipage redirect of	Gelfand-Raikov theorem Gelfand Raikov theorem
is foaf:primaryTopic of	wikipedia-en:Gelfand–Raikov_theorem

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