path: root/pages/operator-algebras/c-star-algebras/states.md

---
title: States
parent: C*-Algebras
grand_parent: Operator Algebras
nav_order: 1
---

# {{ page.title }}

{% definition State, State Space %}
A norm-one [positive linear functional]( {% link pages/operator-algebras/c-star-algebras/positive-linear-functionals.md %} ) 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.
{% enddefinition %}

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

{% corollary %}
A linear functional $\omega$ on a C\*-algebra is a state
if and only if $\omega(\mathbf{1}) = 1 = \norm{\omega}$.
{% endcorollary %}

{% proposition %}
The state space of a C\*-algebra is convex and weak\* compact.
{% endproposition %}

{% 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 $\omega(\mathbf{1}) = 1$.
By the triangle inequality, $\norm{\omega} \le 1$.
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]({% link pages/functional-analysis-basics/banach-alaoglu-theorem.md %}),
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 $\omega_i(\mathbf{1}) = 1$; hence $\omega(x) \ge 0$ for $x \ge 0$ and $\omega(\mathbf{1}) = 1$. Thus $\omega$ is again a state.
This shows that the state space is weak* closed, completing the proof.
{% endproof %}

TODO: state space is nonempty

TODO: pure states