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 %}