PlanetMath: positive linear functional

(more info)

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positive linear functional

(Definition)

Definition

Let $\mathcal{A}$ be a

-algebra and $\phi$ a linear functional on $\mathcal{A}$ .

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

Let $\phi$ be a positive linear functional on $\mathcal{A}$ . Then

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

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

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

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

Examples

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