• Observables $A$ are a Poisson algebra *-algebra. This means a complex algebra $A$ equipped with an antilinear involution $a \mapsto a^{\ast}$ such that $(ab)^{\ast} = b^{\ast} a^{\ast}$ as well as a Lie bracket $\{ -, - \} : A \otimes A \to A$ which is also a derivation with respect to multiplication (and maybe some compatibility between these two structures). Classical examples occur when $A$ is the algebra of complex-valued smooth functions on a symplectic manifold $M$, the involution is pointwise conjugation, and $\{ -, - \}$ is the usual Poisson bracket, and quantum examples occur when $A$ is the algebra of linear operators on a Hilbert space $H$, the involution is adjoint, and $\{ -, - \}$ is the commutator.
• States are *-linear functionals $\mathbb{E} : A \to \mathbb{C}$ on $A$ such that $\mathbb{E}(1) = 1$ and such that $\mathbb{E}(a^{\ast} a) \ge 0$ for all $a$. Classical examples occur when $\mathbb{E}$ is integration against a probability measure on a symplectic manifold $M$. Pure quantum examples occur when $\mathbb{E}(a) = \langle \psi, a \psi \rangle$ for some unit vector $\psi$ in some Hilbert $^{\ast}$-representation of $A$ and you can get more examples by taking linear combinations or more generally integrals of these.

Density operators come into the picture as follows. $A$ is sometimes equipped with a canonical linear functional (which may not be everywhere defined). Classical examples occur when this linear functional is given by integration against Liouville measure on a symplectic manifold $M$, and quantum examples occur when this linear functional is given by trace. If $\text{tr}$ denotes this functional, then you can use it to write down a distinguished class of states of the form

for some $\rho \in A$. This is the density operator of the state. If $\text{tr}$ satisfies a suitable nondegeneracy condition, it can be uniquely recovered from the state. The axioms defining a state require $\rho$ to be self-adjoint and have trace $1$, and it must also have suitable positivity properties (for example in the case of a finite-dimensional matrix algebra it must have non-negative eigenvalues).

• Observables $A$ are a Poisson *-algebra. This means a complex algebra $A$ equipped with an antilinear involution $a \mapsto a^{\ast}$ such that $(ab)^{\ast} = b^{\ast} a^{\ast}$ as well as a Lie bracket $\{ -, - \} : A \otimes A \to A$ which is also a derivation with respect to multiplication (and maybe some compatibility between these two structures). Classical examples occur when $A$ is the algebra of complex-valued smooth functions on a symplectic manifold $M$, the involution is pointwise conjugation, and $\{ -, - \}$ is the usual Poisson bracket, and quantum examples occur when $A$ is the algebra of linear operators on a Hilbert space $H$, the involution is adjoint, and $\{ -, - \}$ is the commutator.
• States are *-linear functionals $\mathbb{E} : A \to \mathbb{C}$ on $A$ such that $\mathbb{E}(1) = 1$ and such that $\mathbb{E}(a^{\ast} a) \ge 0$ for all $a$. Classical examples occur when $\mathbb{E}$ is integration against a probability measure on a symplectic manifold $M$. Pure quantum examples occur when $\mathbb{E}(a) = \langle \psi, a \psi \rangle$ for some unit vector $\psi$ in some Hilbert $^{\ast}$-representation of $A$ and you can get more examples by taking linear combinations or more generally integrals of these.

Density operators come into the picture as follows. $A$ is sometimes equipped with a canonical linear functional (which may not be everywhere defined); a normalized version of the corresponding state (when it exists) should be thought of as the "uniform distribution." Classical examples occur when this linear functional is given by integration against Liouville measure on a symplectic manifold $M$, and quantum examples occur when this linear functional is given by trace. If $\text{tr}$ denotes this functional, then you can use it to write down a distinguished class of states of the form

for some $\rho \in A$. This is the density operator of the state. If $\text{tr}$ satisfies a suitable nondegeneracy condition, $\rho$ can be uniquely recovered from the state it defines. The axioms defining a state require $\rho$ to be self-adjoint and have trace $1$, and it must also have suitable positivity properties (for example in the case of a finite-dimensional matrix algebra it must have non-negative eigenvalues).

• Observables $A$ are a Poisson *-algebra. This means a complex algebra $A$ equipped with an antilinear involution $a \mapsto a^{\ast}$ such that $(ab)^{\ast} = b^{\ast} a^{\ast}$ as well as a Lie bracket $\{ -, - \} : A \otimes A \to A$ which is also a derivation with respect to multiplication (and maybe some compatibility between these two structures). Classical examples occur when $A$ is the algebra of complex-valued smooth functions on a symplectic manifold $M$, the involution is pointwise conjugation, and $\{ -, - \}$ is the usual Poisson bracket, and quantum examples occur when $A$ is the algebra of linear operators on a Hilbert space $H$, the involution is adjoint, and $\{ -, - \}$ is the commutator.
• States are *-linear functionals $\mathbb{E} : A \to \mathbb{C}$ on $A$ such that $\mathbb{E}(1) = 1$ and such that $\mathbb{E}(a^{\ast} a) \ge 0$ for all $a$. Classical examples occur when $\mathbb{E}$ is integration against a probability measure on a symplectic manifold $M$. Pure quantum examples occur when $\mathbb{E}(a) = \langle \psi, a \psi \rangle$ for some unit vector $\psi$ in some Hilbert $^{\ast}$-representation of $A$ and you can get more examples by taking linear combinations or more generally integrals of these.

Density operators come into the picture as follows. $A$ is sometimes equipped with a canonical linear functional (which may not be everywhere defined). Classical examples occur when this linear functional is given by integration against Liouville measure on a symplectic manifold $M$, and quantum examples occur when this linear functional is given by trace. If $\text{tr}$ denotes this functional, then you can use it to write down a distinguished class of states of the form

for some $\rho \in A$. This is the density operator of the state. If $\text{tr}$ satisfies a suitable nondegeneracy condition, it can be uniquely recovered from the state. The axioms defining a state require $\rho$ to be self-adjoint, and it must also satisfy a suitable positivity axiom (for example in the case of a finite-dimensional matrix algebra it must have non-negative eigenvalues).

You are absolutely correct that every state can be described using a vector in a suitably large Hilbert space (here I am using "Hilbert space" in the physicist's sense, which I understand to really be "inner product space"). This is a corollary of a version of the Gelfand-Naimark-Segal construction which is explained in the third post above.

However, it is worth mentioning that there is an intrinsic definition of pure state in the theory of operator algebras: namely, the space of states has a natural convex structure and you can define a pure state to be an extreme point of the space of states.

• Observables $A$ are a Poisson algebra *-algebra. This means a complex algebra $A$ equipped with an antilinear involution $a \mapsto a^{\ast}$ such that $(ab)^{\ast} = b^{\ast} a^{\ast}$ as well as a Lie bracket $\{ -, - \} : A \otimes A \to A$ which is also a derivation with respect to multiplication (and maybe some compatibility between these two structures). Classical examples occur when $A$ is the algebra of complex-valued smooth functions on a symplectic manifold $M$, the involution is pointwise conjugation, and $\{ -, - \}$ is the usual Poisson bracket, and quantum examples occur when $A$ is the algebra of linear operators on a Hilbert space $H$, the involution is adjoint, and $\{ -, - \}$ is the commutator.
• States are *-linear functionals $\mathbb{E} : A \to \mathbb{C}$ on $A$ such that $\mathbb{E}(1) = 1$ and such that $\mathbb{E}(a^{\ast} a) \ge 0$ for all $a$. Classical examples occur when $\mathbb{E}$ is integration against a probability measure on a symplectic manifold $M$. Pure quantum examples occur when $\mathbb{E}(a) = \langle \psi, a \psi \rangle$ for some unit vector $\psi$ in some Hilbert $^{\ast}$-representation of $A$ and you can get more examples by taking linear combinations or more generally integrals of these.

Density operators come into the picture as follows. $A$ is sometimes equipped with a canonical linear functional (which may not be everywhere defined). Classical examples occur when this linear functional is given by integration against Liouville measure on a symplectic manifold $M$, and quantum examples occur when this linear functional is given by trace. If $\text{tr}$ denotes this functional, then you can use it to write down a distinguished class of states of the form

for some $\rho \in A$. This is the density operator of the state. If $\text{tr}$ satisfies a suitable nondegeneracy condition, it can be uniquely recovered from the state. The axioms defining a state require $\rho$ to be self-adjoint and have trace $1$, and it must also have suitable positivity properties (for example in the case of a finite-dimensional matrix algebra it must have non-negative eigenvalues).

You are absolutely correct that every state can be described using a vector in a suitably large Hilbert space (here I am using "Hilbert space" in the physicist's sense, which I understand to really be "inner product space"). This is a corollary of a version of the Gelfand-Naimark-Segal construction which is explained in the third post above.

However, it is worth mentioning that there is an intrinsic definition of pure state in the theory of operator algebras: namely, the space of states has a natural convex structure and you can define a pure state to be an extreme point of the space of states.