GNS Construction for Observable Algebras  

Gell-Mann-Naimark-Segal Construction in General {Mostly from an explanation by Abhay, 1985} > s.a. observable algebras; operator theory.
* Idea: A construction used in algebraic quantum (field) theory, which allows one to construct, given a star-algebra \(\cal A\) and a complex valued "positive" linear functional f defined on it (a state), a Hilbert space \(\cal H\) and a representation of the algebra on the Hilbert space.
$ Construction: If \(\cal A\) is a star-algebra, and f : \(\cal A\) → \(\mathbb C\) a "positive" (i.e., non-negative) linear functional,

for all A ∈ \(\cal A\),   f(A*A) ≥ 0

(we do not require strict positivity because in many useful physical examples we do not have it – f(A) can be the expectation value of A in a given state ψ, for example);
- Define I ⊂ \(\cal A\) as I:= {A ∈ \(\cal A\) | f(A*A) = 0}; This is a subspace of \(\cal A\), in fact it is a left ideal.
- Define V:= \(\cal A\)/I (so that the inner product will be positive);
- Then V can be structured as an inner product space by defining

\(\langle\){A}, {B}\(\rangle\):= f(A*B),   for all A, B ∈ \(\cal A\)

(and, if we complete it in the standard way, it becomes a Hilbert space);
- Also, \(\cal A\) acts naturally an V by B({A}):= {BA} (if we get the Hilbert space by completing V, these operators will in general not be defined on the whole \(\cal H\)).
* Useful result: If \(\cal H\) is a Hilbert space with a cyclic representation of a star-algebra \(\cal A\), i.e., one such that

∃ ψ0 ∈ \(\cal H\) such that (for all ψ ∈ \(\cal H\), ∃ A ∈ \(\cal A\) such that ψ = A ψ0) ,

then this representation is unitarily equivalent to the one obtained by the GNS construction, with f(A):= \(\langle\)ψ0, 0\(\rangle\).
> Online resources: see Wikipedia page.

References > s.a. entanglement; holonomy.
@ General: Reed in(70); Iguri & Castagnino JMP(08)-a0711 [for topological *-algebras]; Chruściński & Marmo OSID(09)-a0810 [geometric description].
@ Deformed algebras: Bordemann & Waldmann CMP(98)qa/96; Waldmann RPMP(01)m.QA/00; Gozzi & Reuter IJMPA(94)ht/03.
@ Other generalizations: Hofmann CMP(98); Naudts & Kuna JPA(01)mp/00 [covariance systems]; Cariñena et al TMP(07)mp [alternative].

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