GNS construction
E924197
concept in functional analysis
concept in operator algebras
mathematical construction
representation-theoretic construction
The GNS construction is a fundamental procedure in functional analysis that represents a C*-algebra as bounded operators on a Hilbert space derived from a given state, providing a bridge between abstract algebraic structures and concrete operator representations.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in functional analysis
ⓘ
concept in operator algebras ⓘ mathematical construction ⓘ representation-theoretic construction ⓘ |
| alsoKnownAs | Gelfand–Naimark–Segal construction NERFINISHED ⓘ |
| appliesTo | positive linear functionals on *-algebras ⓘ |
| category | Hilbert space representation construction ⓘ |
| codomain | Hilbert space representation of a C*-algebra ⓘ |
| defines | cyclic *-representation of a C*-algebra ⓘ |
| domain |
C*-algebra
ⓘ
state on a C*-algebra ⓘ |
| field |
C*-algebra theory
ⓘ
functional analysis ⓘ operator algebras ⓘ von Neumann algebra theory ⓘ |
| generalizationOf | Riesz representation theorem for positive functionals NERFINISHED ⓘ |
| guarantees |
correspondence between states and cyclic representations up to unitary equivalence
ⓘ
every state gives rise to a cyclic representation ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| input |
C*-algebra A
ⓘ
state φ on A ⓘ |
| mathematicalNature | non-constructive up to unitary equivalence ⓘ |
| namedAfter |
Irving E. Segal
NERFINISHED
ⓘ
Israel M. Gelfand NERFINISHED ⓘ Mark A. Naimark NERFINISHED ⓘ |
| output |
Hilbert space H_φ
ⓘ
cyclic vector ξ_φ in H_φ ⓘ representation π_φ of A on H_φ ⓘ |
| property |
construction is unique up to unitary equivalence
ⓘ
representation is cyclic with cyclic vector ξ_φ ⓘ representation is nondegenerate ⓘ |
| purpose |
to associate a cyclic *-representation to a given state
ⓘ
to represent a C*-algebra as bounded operators on a Hilbert space ⓘ |
| relatedTo |
Gelfand–Naimark theorem
NERFINISHED
ⓘ
positive linear functionals ⓘ representation theory of C*-algebras ⓘ states on C*-algebras ⓘ von Neumann algebra representations ⓘ |
| role | provides a bridge between abstract C*-algebras and concrete operator representations ⓘ |
| satisfies | φ(a)=⟨π_φ(a)ξ_φ,ξ_φ⟩ for all a in A ⓘ |
| usedIn |
algebraic quantum field theory
NERFINISHED
ⓘ
classification of representations of C*-algebras ⓘ mathematical quantum mechanics ⓘ theory of KMS states ⓘ |
| uses |
completion of a pre-Hilbert space
ⓘ
inner product induced by the state ⓘ quotient of the algebra by the GNS null space ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.