positive linear functional
0.0.1 Definition
Let $\mathcal{A}$ be a ${C}^{*}$algebra (http://planetmath.org/CAlgebra) and $\varphi $ a linear functional on $\mathcal{A}$.
We say that $\varphi $ is a positive linear functional^{} on $\mathcal{A}$ if $\varphi $ is such that $\varphi (x)\ge 0$ for every $x\ge 0$, i.e. for every positive element^{} $x\in \mathcal{A}$.
0.0.2 Properties
Let $\varphi $ be a positive linear functional on $\mathcal{A}$. Then

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

•
${\varphi ({x}^{*}y)}^{2}\le \varphi ({x}^{*}x)\varphi ({y}^{*}y)$ for every $x,y\in \mathcal{A}$. This is an analog of the CauchySchwartz inequality
Let $\varphi $ be a linear functional on a ${C}^{*}$algebra $\mathcal{A}$ with identity element^{} $e$. Then

•
$\varphi $ is positive if and only if $\varphi $ is bounded^{} (http://planetmath.org/ContinuousLinearMapping) and $\parallel \varphi \parallel =\varphi (e)$.
0.0.3 Examples

•
Let $X$ be a locally compact Hausdorff space^{} and ${C}_{0}(X)$ the ${C}^{*}$algebra of continuous functions $X\u27f6\u2102$ that vanish at infinity. Let $\mu $ be a regular^{} Radon measure^{} on $X$. The linear functional $\varphi $ defined by integration against $\mu $,
$$\varphi (f):={\int}_{X}f\mathit{d}\mu ,f\in {C}_{0}(x)$$ 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  20130322 17:45:05 
Last modified on  20130322 17:45:05 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  11 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 46L05 