In a previous exercise, I have proven that states $\omega$ on the $C^*$-algebra $M_n(\mathbb C)$ correspond to a unique density matrix $\rho$ by the relation $\omega(A) = \mathrm{Tr}(\rho A)$. I was then asked to construct a GNS-triplet for $\omega$, which I did in the following way:
Equipping $M_n(\mathbb C)$ with its Hilbert-Schmidt sesquilinear form and given a state $\omega$ corresponding to a density matrix $\rho$, divide out by the set $$ N :=\{B \in M_n(\mathbb C) \mid \omega(B^*B) = 0\}$$ and we claim that on the space $M_n(\mathbb C)/N$, there exists a representation $\pi_\omega$ with cyclic vector $\xi_\omega$ for this representation such that $\omega(A) = \langle \pi_\omega(A)\xi_\omega, \xi_\omega\rangle$ on this space. If we take a look at the GNS-condition for the representation and cyclic vector and interpret the Hilbert-Schmidt sesquilinear form, we are asked to find $\pi_\omega$ and $\xi_\omega$ such that $$\forall A \in M_n(\mathbb C)/N: \mathrm{Tr}(\rho A) = \mathrm{Tr}(\xi_\omega^* \pi_\omega(A)\xi_\omega)\, .$$ This seems to ask for $\pi_\omega$ to be the left multiplication operator with $A$. Indeed, since $\rho$ was positive, we can set $\xi_\omega := \rho^{1/2}$ and observe that by the `commutative property' of the trace, $$\mathrm{Tr}(\rho A) = \mathrm{Tr}(\xi_\omega A \xi_\omega) = \mathrm{Tr}(\xi_\omega^*\pi_\omega(A)\xi_\omega)\, ,$$ because $\xi_\omega$ is positive and hence self-adjoint.
It remains to show that $\xi_\omega$ is a cyclic vector for the representation $\pi_\omega$. It suffices to prove the following statement: $$\forall B \in M_n(\mathbb C)/N \, \exists A \in M_n(\mathbb C)/N: \pi_\omega(A)\xi_\omega=B\, ,$$ so we need to find $A \in M_n(\mathbb C)$ with $\omega(A^*A) \neq 0$ such that $A\xi_\omega = B$ [argument under construction].
The question I'm struggling with now goes as follows:
Can you generalize the construction you made in the previous exercise to a possible infinite-dimensional setting, i.e. considering for a Hilbert space $\mathcal H$ and a trace-class operator $\rho$ the positive functional $$\omega_\rho: \mathcal B(\mathcal H) \to \mathbb C: A \mapsto \omega_\rho(A) = \mathrm{Tr}(\rho A)\, ?$$
When asked about this problem, my professor gave the following hint:
In the infinite-dimensional setting, only normal states are of the form $\mathrm{Tr}(\rho A)$.
We hadn't seen the notion of a normal state in the course, so I googled its definition (for every monotone increasing net of operators $H_\alpha$ with upper bound $H$, $\omega_\rho(H_\alpha)$ converges to $\omega_\rho(H)$). I don't really see how to tackle the question, or why it is relevant that $\omega_\rho$ should be normal. Can someone get me started on solving the problem and/or proving the hint?
Edit: additionally, I noticed that the argument I gave for cyclicity of $\xi_\omega$ is wrong: it is possible that $(\det \xi_\omega)^2 = \det \rho = 0$, so $\xi_\omega$ is not necessarily invertible. Therefore, I ask another question: how do I prove that $\xi_\omega$ is a cyclic vector for $\pi_\omega$ on $M_n(\mathbb C)/N$? Assuming that $\omega(A^*A) \neq 0$ is equivalent to saying that $\mathrm{Tr}(\xi_\omega A) \neq 0$.
I was hesitant to tag this post as homework since this is not an assignment, but just an exercise in the course notes of our Operator Algebras course. Please let me know if I violated any rules concerning posting on StackExchange, as this is my first post. Thanks in advance for your answers!
I contacted my professor with this question and am adding his response here should anyone need it in the future:
For the finite-dimensional case, my representation and cyclic vector were correct. In case $\det \xi_\omega = 0$, $\xi_\omega$ is not cyclic on $M_n(\mathbb C)$, but it is cyclic on $$\{ A\xi_\omega \mid A \in M_n(\mathbb C)\}$$ per construction. (My professor said "if $\xi_\omega$ is not invertible, then indeed you do not retrieve all of $M_n(\mathbb C)$" and I'm a bit puzzled: is there some explicit way to describe this set if $\xi_\omega$ is not invertible that I'm not seeing or am I being paranoid?)
In the infinite-dimensional setting, as Michael noted there is the issue is that not all operators are associated to a trace-class operator. The exercise my professor meant to give was to construct GNS-triplets for those states that are, so I misunderstood his question as I was confused by the hint. As Hilbert space, one should take the Hilbert-Schmidt operators with not-necessarily-cyclic vector $\rho^{1/2}$: if $\rho$ admits zero as eigenvalue, one should again 'restrict' to a space on which $\rho^{1/2}$ is cyclic per construction.