PositiveLinearFunctional 2008-01-19 18:31:54 2008-04-15 15:36:40 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \subsubsection{Definition} Let $\mathcal{A}$ be a \PMlinkname{$C^*$-algebra}{CAlgebra} and $\phi$ a linear functional on $\mathcal{A}$. We say that $\phi$ is a {\bf 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 $x \in \mathcal{A}$. \subsubsection{Properties} Let $\phi$ be a positive linear functional on $\mathcal{A}$. Then \begin{itemize} \item $\phi(x^*) = \overline{\phi(x)}\;\;$ for every $x \in \mathcal{A}$. \end{itemize} \begin{itemize} \item $|\phi(x^*y)|^2 \leq \phi(x^*x)\phi(y^*y)\;\;$ for every $x, y \in \mathcal{A}$. This is an analog of the Cauchy-Schwartz inequality \end{itemize} Let $\phi$ be a linear functional on a $C^*$-algebra $\mathcal{A}$ with identity element $e$. Then \begin{itemize} \item $\phi$ is positive if and only if $\phi$ is \PMlinkname{bounded}{ContinuousLinearMapping} and $\|\phi\|= \phi(e)$. \end{itemize} \subsubsection{Examples} \begin{itemize} \item 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 $\mu$, \begin{displaymath} \phi(f) := \int_X f\;d\mu\;, \qquad\qquad f \in C_0(x) \end{displaymath} is a positive linear functional on $C_0(X)$. In fact, by the \PMlinkname{Riesz representation theorem}{RieszRepresentationTheoremOfLinearFunctionalsOnFunctionSpaces}, all positive linear functionals of $C_0(X)$ are of this form. \end{itemize}