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

---
 .../c-star-algebras/positive-linear-functionals.md | 40 ++++++++++------------
 1 file changed, 18 insertions(+), 22 deletions(-)

(limited to 'pages/operator-algebras/c-star-algebras/positive-linear-functionals.md')

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 %}
-- 
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