Definition

We say that $\phi$ is a positive linear functional on $\mathcal{A}$ if $\phi$ is such that $\phi(x)\geq 0$ for every $x \geq 0$ , i.e. for every positive element $x \in \mathcal{A}$ .

Properties

• $\phi(x^*) = \overline{\phi(x)}\;\;$ for every $x \in \mathcal{A}$ .

• $|\phi(x^*y)|^2 \leq \phi(x^*x)\phi(y^*y)\;\;$ for every $x, y \in \mathcal{A}$ . This is an analog of the Cauchy-Schwartz inequality

Let $\phi$ be a linear functional on a $C^*$ -algebra $\mathcal{A}$ with identity element $e$ . Then

• $\phi$ is positive if and only if $\phi$ is bounded and $\|\phi\|= \phi(e)$ .

Examples

• Let $X$ be a locally compact Hausdorff space and $C_0(X)$ the $C^*$ -algebra of continuous functions $X \longrightarrow \mathbb{C}$ that vanish at infinity. Let $\mu$ be a regular Radon measure on $X$ . The linear functional $\phi$ defined by integration against $\mu$ ,
  
  is a positive linear functional on $C_0(X)$ . In fact, by the Riesz representation theorem, all positive linear functionals of $C_0(X)$ are of this form.