positive linear functional

0.0.1 Definition

Let $\mathcal{A}$ be a $C^{*}$ -algebra (http://planetmath.org/CAlgebra) 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 $x\in\mathcal{A}$ .

0.0.2 Properties

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

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

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$|\phi(x^{*}y)|^{2}\leq\phi(x^{*}x)\phi(y^{*}y)\;\;$ 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

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$\phi$ is positive if and only if $\phi$ is bounded (http://planetmath.org/ContinuousLinearMapping) and $\|\phi\|=\phi(e)$ .

0.0.3 Examples

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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 $\mu$ ,

$\phi(f):=\int_{X}f\;d\mu\;,\qquad\qquad f\in C_{0}(x)$

is a positive linear functional on $C_{0}(X)$ . In fact, by the Riesz representation theorem (http://planetmath.org/RieszRepresentationTheoremOfLinearFunctionalsOnFunctionSpaces), all positive linear functionals of $C_{0}(X)$ are of this form.

Title	positive linear functional
Canonical name	PositiveLinearFunctional
Date of creation	2013-03-22 17:45:05
Last modified on	2013-03-22 17:45:05
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	11
Author	asteroid (17536)
Entry type	Definition
Classification	msc 46L05