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$$\rangle$$.