I have a question concerning $n \times n$ matrices.
Denote by $M_n(\mathbb{C})$ the algebra of all $n \times n$ matrices with complex entries. Let $\phi$ be a state on $M_n(\mathbb{C})$ i.e. a linear functional $\phi \colon M_n(\mathbb{C}) \rightarrow \mathbb{C}$ such that $\phi(Id)=1$ and $\phi(A^*A) \geq 0$, for each $A \in M_n(\mathbb{C})$.
Show that $\phi$ must be of the form $\phi(A) = \text{tr}(\rho A)$, where $\rho$ is a positive-definite matrix such that $\mbox{tr}(\rho)=1$.
Thank you for any help.