2 files changed, 36 insertions, 34 deletions
diff --git a/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md
index 05b1d4f..ea15f87 100644
--- a/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md
+++ b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md
@@ -3,37 +3,33 @@ title: Positive Linear Functionals
 parent: C*-Algebras
 grand_parent: Operator Algebras
 nav_order: 1
-# cspell:words
 ---
 
 # {{ page.title }}
 
 all algebra are assumed to be unital
 
-{: .definition-title }
-> Hermitian Functional, Positive Functional, State
->
-> A linear functional on a $C^*$-algebra $\mathcal{A}$ is said to be
-> 
-> - *Hermitian* if $\phi(x*) = \overline{\phi(x)}$ for all $x \in \mathcal{A}$.
-> - *positive* if $\phi(x) \ge 0$ for all $x \ge 0$.
-> - a *state* if $\phi$ is positive and $\phi(\mathbf{1}) = 1$.
->
-
-{: .definition-title }
-> State
->
-> A norm-one positive linear functional on a $C^*$-algebra is called a *state*.
-
-{: .definition-title }
-> State Space
->
-> The *state space* of a $C^*$-algebra $\mathcal{A}$, denoted by $S(\mathcal{A})$, is the set of all states of $\mathcal{A}$.
+{% definition Hermitian Functional, Positive Functional, State %}
+A linear functional on a $C^*$-algebra $\mathcal{A}$ is said to be
+
+- *Hermitian* if $\phi(x*) = \overline{\phi(x)}$ for all $x \in \mathcal{A}$.
+- *positive* if $\phi(x) \ge 0$ for all $x \ge 0$.
+- a *state* if $\phi$ is positive and $\phi(\mathbf{1}) = 1$.
+{% enddefinition %}
+
+{% definition State %}
+A norm-one positive linear functional on a $C^*$-algebra is called a *state*.
+{% enddefinition %}
+
+{% definition State Space %}
+The *state space* of a $C^*$-algebra $\mathcal{A}$, denoted by $S(\mathcal{A})$, is the set of all states of $\mathcal{A}$.
+{% enddefinition %}
 
 Note that $S(\mathcal{A})$ is a subset of the unit ball in the dual space of $\mathcal{A}$.
 
-{: .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 %}
 {% endproof %}
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
+