1 files changed, 39 insertions, 0 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
new file mode 100644
index 0000000..05b1d4f
--- /dev/null
+++ b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md
@@ -0,0 +1,39 @@
+---
+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 %}