Let $\mathcal{A}$ be a noncommutative $C^*$-algebra and $PS(\mathcal{A})$ be the set of its pure states.

**Question 1.** Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\mathcal{A}$ is commutative, $PS(\mathcal{A})$ is easily seen to be linearly independent (I learned this from Bernard Russo).)

**Question 2.** If $\mathcal{A}$ is unital, is $PS(\mathcal{A})$ always compact in the weak* topology? If not, is there any topology that makes $PS(\mathcal{A})$ compact?

Bloch sphere). $\endgroup$ – Igor Khavkine Jul 24 '16 at 13:44