From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Wed, 14 Feb 2024 07:24:38 +0100
Subject: Update

---
 pages/operator-algebras/c-star-algebras/states.md | 30 ++++++++++++++---------
 1 file changed, 18 insertions(+), 12 deletions(-)

(limited to 'pages/operator-algebras/c-star-algebras/states.md')

diff --git a/pages/operator-algebras/c-star-algebras/states.md b/pages/operator-algebras/c-star-algebras/states.md
index 619bc9a..29cf5f5 100644
--- a/pages/operator-algebras/c-star-algebras/states.md
+++ b/pages/operator-algebras/c-star-algebras/states.md
@@ -3,27 +3,28 @@ title: States
 parent: C*-Algebras
 grand_parent: Operator Algebras
 nav_order: 1
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .definition-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.
+{% 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 unit ball in the dual space of $\mathcal{A}$.
+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 $\omega(\mathbf{1}) = 1 = \norm{\omega}$.
+{% 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.
+{% 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 $\omega = (1-t)\omega_0 + t\omega_1$.
@@ -41,3 +42,8 @@ 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
+
-- 
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