in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$

Now, of course there is also in classical physics and quantum mechanics the definition of a state.

In classical physics this is either a point in phase space or more generally a probability measure on this space.

In quantum mechanics this is either a wavefunction or a density matrix.

Now there are basically two interesting examples of $C^*-$ algebras I would say: $L(H)$ the space of bounded operators on some Hilbert space $H$ (a non-comm. $C^*-$ algebra or $C_0(X)$ the space of $C_0-$ functions on some paracompact Hausdorff-space (a commutative one).

Obviously, if $X$ is some compact subset of $\mathbb{R}^n,$ then $C(X)$ is a $C^*-$ algebra with unit element and dirac measures on $X$ and more generally probability measures are indeed states as we defined them in the functional analytic sense.

Moreover, if we work on some Hilbert space $H$ then the density matrices $\rho$ define functionals $l:L(H) \rightarrow \mathbb{R}, T \mapsto tr(\rho T).$ So these are also states in the sense of functional analysis.

But this made me think whether

1.) Every state in the sense of functional analysis can be interpreted as a physical state?

2.) Where does the interpretation come from that the commutative $C^*-$algebra cooresponds to classical mechanics and the non-commutative one to quantum mechanics? Is there any deep interpretation of this fact? (besides the fact that non-commutativity is known to be an issue for QM?)Cause this seems to be much deeper here as this is the only distinguishing fact between the two in this setup.

in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$

Now, of course there is also in classical physics and quantum mechanics the definition of a state.

In classical physics this is either a point in phase space or more generally a probability measure on this space.

In quantum mechanics this is either a wavefunction or a density matrix.

Now there are basically two interesting examples of $C^*-$ algebras I would say: $L(H)$ the space of bounded operators on some Hilbert space $H$ (a non-comm. $C^*-$ algebra or $C_0(X)$ the space of $C_0-$ functions on some locally compact Hausdorff-space (a commutative one).

Obviously, if $X$ is some compact subset of $\mathbb{R}^n,$ then $C(X)$ is a $C^*-$ algebra with unit element and dirac measures on $X$ and more generally probability measures are indeed states as we defined them in the functional analysis sense.

Moreover, if we work on some Hilbert space $H$ then the density matrices $\rho$ define functionals $l:L(H) \rightarrow \mathbb{R}, T \mapsto tr(\rho T).$ So these are also states in the sense of functional analysis.

But this made me think whether

1.) Every state in the sense of functional analysis can be interpreted as a physical state?

2.) Where does the interpretation come from that the commutative $C^*-$algebra cooresponds to classical mechanics and the non-commutative one to quantum mechanics? Is there any deep interpretation of this fact? (besides the fact that non-commutativity is known to be an issue for QM?)Cause this seems to be much deeper here as this is the only distinguishing fact between the two in this setup.