Positive Linear Functionals

all algebra are assumed to be unital

Definition (Hermitian Functional, Positive Functional, State)

A linear functional on a $C^{\ast }$-algebra $\mathcal{A}$ is said to be

• Hermitian if $\varphi (x\ast )=\overline{\varphi (x)}$ for all $x\in \mathcal{A}$.
• positive if $\varphi (x)\ge 0$ for all $x\ge 0$.
• a state if $\varphi$ is positive and $\varphi (1)=1$.

Definition (State)

A norm-one positive linear functional on a $C^{\ast }$-algebra is called a state.

Definition (State Space)

The state space of a $C^{\ast }$-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 $\mathcal{A}$.

The state space of a $C^{\ast }$-algebra is convex and weak* compact.