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 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
on *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

∃ *ψ*_{0} ∈ \(\cal H\) such
that (for all *ψ* ∈ \(\cal H\),
∃ *A* ∈ \(\cal A\) such
that *ψ* = *A* *ψ*_{0}) ,

then this representation is unitarily equivalent to the one obtained by the
GNS construction, with *f*(*A*):= \(\langle\)*ψ*_{0},
*A**ψ*_{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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