Let $\mathcal{A}$ be a $C^*$ -algebra and $x \in \mathcal{A}$ .

Let $S(\mathcal{A})$ and $P(\mathcal{A})$ denote the state space and the pure state space of $\mathcal{A}$ , respectively.

States

Theorem 1- We have that

• $S(\mathcal{A})$ separates points, i.e. $x= 0$ if and only if $\phi(x) = 0$ for all $\phi \in S(\mathcal{A})$ .
• $x$ is self-adjoint if and only if $\phi(x) \in \mathbb{R}$ for all $\phi \in S(\mathcal{A})$ .
• $x$ is positive if and only if $\phi(x) \geq 0$ for all $\phi \in S(\mathcal{A})$ .
• If $x$ is normal, then $\phi(x) = \|x\|$ for some $\phi \in S(\mathcal{A})$ .

Pure states

The pure state space is also sufficiently large to satisfy the properties of Theorem 1. Hence, we can replace $S(\mathcal{A})$ by $P(\mathcal{A})$ , or by any other family of linear functionals $F$ such that $P(\mathcal{A}) \subset F \subset S(\mathcal{A})$ , in the previous result.

Theorem 2 - We have that

• $P(\mathcal{A})$ separates points, i.e. $x= 0$ if and only if $\phi(x) = 0$ for all $\phi \in P(\mathcal{A})$ .
• $x$ is self-adjoint if and only if $\phi(x) \in \mathbb{R}$ for all $\phi \in P(\mathcal{A})$ .
• $x$ is positive if and only if $\phi(x) \geq 0$ for all $\phi \in P(\mathcal{A})$ .
• If $x$ is normal, then $\phi(x) = \|x\|$ for some $\phi \in P(\mathcal{A})$ .

Proposition - Every multiplicative linear functional on $\mathcal{A}$ is a pure state.