Here is another approach. Instead of defining a state space $L^2(\text{something})$ and then defining observables as the self-adjoint operators on that space, you can go the other way around:
Choose an abstract (unital, $C*$)-algebra$A$ of observables, and then define states as a certain class of linear functionals $A\to\mathbb{C}$.It is important that these states are not vectors in any Hilbert space yet.
After choosing a fixed state $\rho$, you can always find a concrete representation of $A$ (meaning a representation as operators on a Hilbert space), such that $\rho$ is just a vector in that Hilbert space (GNS construction). But this is not unique. Different $\rho$ can give you different representations with different Hilbert spaces. While one representation contains many states (both vector states and other "mixed" states), typically it is impossible to captured all states in a single representation.
In this view, the (algebra of) observables is the fundamental object, notany particular Hilbert space. This approach is common for example in "algebraic quantum field theory".
