pages/operator-algebras/c-star-algebras/positive-linear-functionals.md


1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35

---
title: Positive Linear Functionals
parent: C*-Algebras
grand_parent: Operator Algebras
nav_order: 1
---

# {{ page.title }}

all algebra are assumed to be unital

{% 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.
{% endproposition %}

{% proof %}
{% endproof %}