# Positive Linear Functionals

all algebra are assumed to be unital

Definition (Hermitian Functional, Positive Functional, State)

A linear functional on a $C_{∗}$-algebra $A$ is said to be

Hermitianif $ϕ(x∗)=ϕ(x) $ for all $x∈A$.positiveif $ϕ(x)≥0$ for all $x≥0$.- a
stateif $ϕ$ is positive and $ϕ(1)=1$.

Definition (State)

A norm-one positive linear functional on a $C_{∗}$-algebra is called a

state.

Definition (State Space)

The

state spaceof a $C_{∗}$-algebra $A$, denoted by $S(A)$, is the set of all states of $A$.

Note that $S(A)$ is a subset of the unit ball in the dual space of $A$.

The state space of a $C_{∗}$-algebra is convex and weak* compact.