I was recently made aware (thanks to the answers on Why does Riesz's Representation Theorem apply in quantum mechanics?) that the $C^*$ algebra approach and the Hilbert space approach to quantum mechanics don't perfectly align, and the relationship is somewhat subtle.

### Question

If I start with a the Borel algebra on a locally compact Hausdorff space $X$, how do I get a correspondence between probability measures and states (that potentially satisfy a certain property) in the Hilbert space approach?

### Details

In the $C^*$ algebra approach, this is easy: the states of the algebra $C_c(X)$ of compactly supported complex-valued continuous functions on $X$ are exactly the ones given by Radon probability measures. (The story is also wrapped nicely in a bow by saying that commutative $C^*$ algebras always arise in this way.)

But in the Hilbert space approach, I only know how to do this for the case that $X$ is finite and discrete... In that case, let $H=\mathbb{R}^{|X|}$, and then the states (in the Hilbert approach sense, meaning positive semi-definite trace 1 operators) whose eigenbasis is the standard basis are in correspondence with probability measures on $X$.

The story seems more difficult in the case that $X$ is non-finite and non-discrete. Is it true that there is a procedure for this $X$ to induce a Hilbert space in such a way that states that satisfy a certain property correspond to probability measures on $X$? How should one define this $H$? What should its inner product be, and how would it reflect the topology of $X$? What property should we demand of the states (corresponding to the demand that their eigenbasis is the standard basis in the case that $X$ is finite) so that this correspondence holds?

### Clarification (added later)

I am aware that you can do the following construction: Embed $C_c(X)$ into the algebra of bounded operators on some Hilbert space $H$ via the GNS construction. This is not what I am looking for in this question: it doesn't generalize the discrete finite $X$ case as I described it above. I described a correspondence between probability measures on $X$ and trace 1 diagonal matrices with non-negative entries in $End(\mathbb{R}^{|X|})$. But with the GNS construction, $C_c(X)$ is itself $\mathbb{R}^{|X|}$, but then the construction embeds it into the algebra of bounded operators on a much bigger Hilbert space (and the states of $C_c(X)$ are all pure states in this algebra of bounded operators). So I would not consider this as being naturally a Hilbert approach state, but rather degenerating to the $C^*$ algebra approach.

In order to be a little more concrete, let me try to ask about the following example: Let $X=\mathbb{R}$. Under whatever framework you suggest, even if it goes through the GNS construction, can you describe the standard normal distribution explicitly as some specific operator?

complexHilbert spaces in these contexts... I apologize in advance if I am missing something... $\endgroup$1more comment