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?
Thanks!