# Pure States of Multi-Matrix Algebras

Let $$A$$ be a finite dimensional $$\mathrm{C}^*$$-algebra $$A=\bigoplus_{j=1}^mM_{n_j}(\mathbb{C}).$$

Question 1: Is it true that every state on $$A$$ is of the form: $$\rho=\bigoplus_{j=1}^m \rho_{j},$$

where each $$\rho_j$$ is a positive functional on $$M_{n_j}(\mathbb{C})$$?

Isn't this obviously true... or am I missing something? More importantly for me:

Question 2: Moreover, either as a corollary or a separate result, is each pure state of $$A$$ equal to zero on every factor except for one? i.e. each pure state on $$A$$ arises from a pure state on one of the matrix factors.

For question 2, yes. For question 1, almost yes, every state is of the form $$\rho=\bigoplus^{m}_{j=1}\lambda_{j}\rho_{j}$$ where $$\rho_{j}$$ is a state on $$M_{n_{j}}(\mathbb{C})$$ and $$\lambda_{j}\in\mathbb{R}_{\geq 0}$$ such that $$\sum^{m}_{j=1}\lambda_{j}=1$$. This to ensure that $$\rho(I_{A})=\sum^{m}_{j=1}\lambda_{j}\rho_{j}(I_{n_{j}})=\sum^{m}_{j=1}\lambda_{j}\rho_{j}=1.$$
• Thank you. In fairness I knew it was a convex combination so I wrote that $\rho_j$ were positive functionals. – JP McCarthy Jun 25 at 13:30