PropertiesOfStates 2008-01-23 17:57:56 2008-01-23 18:01:26 Theorem state % 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 Let $\mathcal{A}$ be a \PMlinkname{$C^*$-algebra}{CAlgebra} and $x \in \mathcal{A}$. Let $S(\mathcal{A})$ and $P(\mathcal{A})$ denote the \PMlinkname{state}{State} space and the pure state space of $\mathcal{A}$, respectively. \subsection{States} The \PMlinkescapetext{state} space is sufficiently large to reveal many \PMlinkescapetext{properties} of elements of a $C^*$-algebra. {\bf Theorem 1-} We have that \begin{itemize} \item $S(\mathcal{A})$ separates points, i.e. $x= 0$ if and only if $\phi(x) = 0$ for all $\phi \in S(\mathcal{A})$. \item $x$ is \PMlinkname{self-adjoint}{InvolutaryRing} if and only if $\phi(x) \in \mathbb{R}$ for all $\phi \in S(\mathcal{A})$. \item $x$ is positive if and only if $\phi(x) \geq 0$ for all $\phi \in S(\mathcal{A})$. \item If $x$ is \PMlinkname{normal}{InvolutaryRing}, then $\phi(x) = \|x\|$ for some $\phi \in S(\mathcal{A})$. \end{itemize} \subsection{Pure states} The pure state space is also sufficiently large to \PMlinkescapetext{satisfy} the \PMlinkescapetext{properties} of Theorem 1. Hence, we can replace $S(\mathcal{A})$ by $P(\mathcal{A})$, or by any other family of linear functionals $F$ such that $P(\mathcal{A}) \subset F \subset S(\mathcal{A})$, in the previous result. {\bf Theorem 2 -} We have that \begin{itemize} \item $P(\mathcal{A})$ separates points, i.e. $x= 0$ if and only if $\phi(x) = 0$ for all $\phi \in P(\mathcal{A})$. \item $x$ is \PMlinkescapetext{self-adjoint} if and only if $\phi(x) \in \mathbb{R}$ for all $\phi \in P(\mathcal{A})$. \item $x$ is positive if and only if $\phi(x) \geq 0$ for all $\phi \in P(\mathcal{A})$. \item If $x$ is \PMlinkescapetext{normal}, then $\phi(x) = \|x\|$ for some $\phi \in P(\mathcal{A})$. \end{itemize} {\bf \PMlinkescapetext{Proposition} -} Every multiplicative linear functional on $\mathcal{A}$ is a pure state.