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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.

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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 $$\rho(I_{A})=\sum^{m}_{j=1}\lambda_{j}\rho_{j}(I_{n_{j}})=\sum^{m}_{j=1}\lambda_{j}\rho_{j}=1.$$

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  • $\begingroup$ Thank you. In fairness I knew it was a convex combination so I wrote that $\rho_j$ were positive functionals. $\endgroup$ – JP McCarthy Jun 25 at 13:30
  • $\begingroup$ True, but taking arbitrary positive functionals would not automatically yield a state. $\endgroup$ – Floris Claassens Jun 25 at 13:33

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