States

Definition (State, State Space)

A normalized positive linear functional on a C*-algebra is called a state.
The state space $S(\mathcal{A})$ of a C*-algebra $\mathcal{A}$ is the set of all its states.

Note that $S(\mathcal{A})$ is a subset of the closed unit ball in the dual space of $\mathcal{A}$.

A linear functional $\omega$ on a C*-algebra is a state if and only if $\omega (1)=1=\Vert \omega \Vert$.

The state space of a C*-algebra is convex and weak* compact.

Proof   Let $\mathcal{A}$ be a C*-algebra and let $S(\mathcal{A})$ be its state space. First, we show convexity. Let ${\omega }_0,{\omega }_1$ be states on $\mathcal{A}$ and let $t\in (0,1)$. Consider the convex combination $\omega =(1-t){\omega }_0+t{\omega }_1$. Clearly, $\omega$ is linear and $\omega (1)=1$. By the triangle inequality, $\Vert \omega \Vert \le 1$. It follows from the lemma above that $\omega$ lies in $S(\mathcal{A})$. This proves that $S(\mathcal{A})$ is convex.

Next we show weak* compactness. Since $S(\mathcal{A})$ is contained in the closed unit ball in the dual of $\mathcal{A}$, which is weak* compact by the Banach–Alaoglu Theorem, it will suffice to show that $S(\mathcal{A})$ is weak* closed. Let $({\omega }_i)$ be a net of states that weak* converges to some bounded linear functional $\omega$ on $\mathcal{A}$. This means that ${\omega }_i(x)\to \omega (x)$ for every $x\in \mathcal{A}$. For all $i$ we have ${\omega }_i(x)\ge 0$ for $x\ge 0$ and ${\omega }_i(1)=1$; hence $\omega (x)\ge 0$ for $x\ge 0$ and $\omega (1)=1$. Thus, $\omega$ is again a state. This shows that the state space is weak* closed, completing the proof. $\square$

TODO: state space is nonempty

Definition (Pure State)

We say that a state of a C*-algebra $\mathcal{A}$ is pure if it is an extreme point of the state space $S(\mathcal{A})$.
The set of pure states of $\mathcal{A}$ is denoted by $P(\mathcal{A})$.