From 777f9d3fd8caf56e6bc6999a4b05379307d0733f Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Tue, 12 Sep 2023 07:36:33 +0200
Subject: Initial commit

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 .../c-star-algebras/positive-linear-functionals.md | 39 ++++++++++++++++++++++
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 create mode 100644 pages/operator-algebras/c-star-algebras/positive-linear-functionals.md

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+---
+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 $\mathcal{A}$.
+
+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.
+
+{% proof %}
+{% endproof %}
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