What is the relation between Hilbert space constructed from the GNS construction and the standard Hilbert space-state?

Question

I recently started reading Algebraic quantum mechanics. So I have no knowledge of the subject. In the GNS construction we construct the Hilbert space of states as follows,

1. We endow the algebra of observables $\mathfrak{A}$ with an inner product using the state $\omega$ which is a linear functional on the space of observables. This inner product may be degenerate. (non zero element might have zero norm in this inner product)
2. Remove these null vectors by quotienting the null space $\mathfrak{N}$ hence giving a positive definite inner product on $\mathfrak{A}/\mathfrak{N}$.
3. Completing this space we get a Hilbert space. The algebra of observables acts naturally on this Hilbert space.

How is the state used to give an inner product? In this case how does the operators corresponding to the observables operate on this Hilbert space? How is it related to the standard Hilbert space state formulism?

Answer 1

Each element of the Hilbert space is a Cauchy sequence of equivalence classes of operators.

So $\vec v=([a_1],[a_2],\dots)$ where $[a]=\{A\in\mathfrak A: \omega(A-a)=0\}$ and where $(\mathcal C_1,\mathcal C_2,\dots)$ is the specific function (sequence) that maps $n\mapsto \mathcal C_n$ and where the sequence is Cauchy.

So now you have an operator $B$ and a vector $\vec v=([a_1],[a_2],\dots)$ and the obvious operation is $B\vec v=([Ba_1],[Ba_2],\dots)$ but you need to show it is well defined. Firstly that it didn't depend on the representative of the equivalence class, that $\omega(a-b)=0$ implies $\omega(Ba-Bb)=0$ and secondly that$([Ba_1],[Ba_2],\dots)$ is Cauchy. Though if it isn't, then you could just say that $\vec v=([a_1],[a_2],\dots)$ isn't in the domain of the unbounded operator.

How does it correspond to observation?

The same as always, the measurement sends a vector to its orthogonal projection onto an eigenspace. The relative frequency of getting a particular eigenspace is the ratio of the squared norm before and after the projection.

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Technically the space of Cauchy sequences still won't be a Hilbert space becasue we didn't finish the completion. Given two Cauchy sequences $([a_1],[a_2],\dots)$ and $([b_1],[b_2],\dots)$ we identify them with the same vector in the Hilbert space if $([a_1-b_1],[a_2-b_2],\dots)$ has zero as a limit (and we have to show that definition is well defined).

So a vector in the Hilbert space is a set of Cauchy sequences. Each Cauchy sequence has values which themselves are sets of operators.

So $\vec v=[([a_1],[a_2],\dots)]$ where the outer $[\,]$ identified two Cauchy sequences if the difference has zero as a limit. And the inner $[\,]$ identifies two operators is their difference has zero as the resukt of $\omega$ and the $(\,)$ just denotes a sequence by listing the values of the sequence in order (and I might be using the axiom of choice in my choice of notation by denoting each equivalence class by a representative).

This means the operator also has to be shown to be well defined on two Cauchy sequence that are identified.