# 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 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

• $\phi(x^{*})=\overline{\phi(x)}\;\;$ 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 (http://planetmath.org/ContinuousLinearMapping) and $\|\phi\|=\phi(e)$.

## 0.0.3 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 $\mu$,

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

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 PositiveLinearFunctional 2013-03-22 17:45:05 2013-03-22 17:45:05 asteroid (17536) asteroid (17536) 11 asteroid (17536) Definition msc 46L05