Gelfand–Naimark–Segal construction

E924198

C*-algebra representation construction construction in functional analysis representation theorem

The Gelfand–Naimark–Segal construction is a fundamental procedure in functional analysis that represents abstract C*-algebras as concrete operators on a Hilbert space via states, forming the basis of the GNS representation.

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Observed surface forms (1)

Surface form	Occurrences
Gårding–Wightman reconstruction theorem	1

Statements (48)

Predicate	Object
instanceOf	C*-algebra representation construction ⓘ construction in functional analysis ⓘ representation theorem ⓘ
alsoKnownAs	GNS construction NERFINISHED ⓘ GNS representation construction NERFINISHED ⓘ
appliesTo	C-algebras ⓘ states on C-algebras ⓘ
assumption	state is a positive normalized linear functional ⓘ
coreConcept	cyclic representation ⓘ positive linear functional ⓘ realization of states as vector states ⓘ representation of C*-algebras by bounded operators on a Hilbert space ⓘ
defines	GNS representation NERFINISHED ⓘ
field	C*-algebra theory ⓘ functional analysis ⓘ operator algebras ⓘ
generalizationOf	representation of commutative C*-algebras by multiplication operators ⓘ
guarantees	existence of a cyclic representation for every state ⓘ uniqueness of the GNS representation up to unitary equivalence ⓘ
historicalPeriod	20th century mathematics ⓘ
importance	basis of the GNS representation used in quantum theory ⓘ fundamental tool in the theory of C*-algebras ⓘ
input	C-algebra NERFINISHED ⓘ state on a C-algebra ⓘ
mathematicalDomain	functional analysis ⓘ operator theory ⓘ
namedAfter	Irving Segal NERFINISHED ⓘ Israel Gelfand NERFINISHED ⓘ Mark Naimark NERFINISHED ⓘ
output	Hilbert space NERFINISHED ⓘ cyclic representation of a C*-algebra ⓘ cyclic vector ⓘ
property	functorial up to unitary equivalence with respect to *-homomorphisms preserving states ⓘ
relatedTo	Gelfand–Naimark theorem NERFINISHED ⓘ Riesz representation theorem NERFINISHED ⓘ Stinespring dilation theorem NERFINISHED ⓘ representation theory of C*-algebras ⓘ von Neumann algebras NERFINISHED ⓘ
role	identifies states with vector states in a Hilbert space representation ⓘ provides canonical representation associated to a state ⓘ represents abstract C*-algebras as concrete operator algebras on Hilbert spaces ⓘ
usedIn	algebraic quantum field theory ⓘ mathematical quantum mechanics ⓘ quantum statistical mechanics ⓘ
usesConcept	Hilbert space completion ⓘ bounded *-representation ⓘ inner product induced by a state ⓘ quotient by null space ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Gelfand–Naimark theorem → hasVariant → Gelfand–Naimark–Segal construction ⓘ

Wightman correlation functions → relatedConcept → Gelfand–Naimark–Segal construction ⓘ

this entity surface form: Gårding–Wightman reconstruction theorem