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I am reading some papers from the 70s on operator theory. I come across the term 'vector state' of a $C^*$-algebra quite often. It is a little bit confusing. Wikipedia redirects to quantum state vector which I found irrelevant.

So could someone give me a definition of a 'vector state'?

In particular, I want to make sense of the following from one of arveson's papers:

If $M_n$ is the algebra of n-n matrices, $\mathcal{A}$ is a $C^*$-algebra in $B(\mathcal{H})$, $\epsilon_j$'s are basis for $\mathbb{C}^n$, then every vector state $\omega$ of $M_n\otimes\mathcal{A}$ is of the form \begin{equation} \omega((A_{ij}))=\sum_{i,j}(V^*A_{ij}V\epsilon_j,\epsilon_i),\end{equation}where $V$ is an operator from $\mathbb{C}^n$ to $\mathcal{H}$.

This somehow resembles the positive linear functionals or states on $C^*$-algebras, but I am not quite sure. Is that the formulae for states on tensors of algebras?


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up vote 2 down vote accepted

If $\xi$ is a unit vector on a Hilbert space then the linear functional $w_\xi: B(H)\to \mathbb C$ given by $w_\xi(a)=\langle a\xi,\xi\rangle$ is positive and $w_\xi(1)=1$ so it is a state and it is called a vector state.

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So you mean states and vector states are the same thing? But in Glimm's paper: A Stone-Weierstrass theorem for $C^*$-algebras, he took great effort to show that any state that annihilates the compact operators is a limit of vector states, which does not make sense if vector states and states are the same thing. – Hui Yu Jul 22 '12 at 15:41
@HuiYu No, they are not the same. A vector space is a state that is of the form $w_\xi$ for some unit vector $\xi$. Not all states are of this form. – azarel Jul 22 '12 at 17:19

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