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\DeclareMathOperator*{\sotstarlim}{\textsc{sot*}--lim} \DeclareMathOperator*{\sotlim}{\textsc{sot}--lim} \newcommand{\wotsum}{\textsc{wot--}\!\!\sum} \newcommand{\sotsum}{\textsc{sot--}\!\!\sum} \newcommand{\wotto}{\overset{\textsc{wot}}\longrightarrow} \newcommand{\sotto}{\overset{\textsc{sot}}\longrightarrow} \newcommand{\ra}{\rightarrow} \newcommand{\la}{\leftarrow} \newcounter{exerci}[section] \renewcommand{\theexerci}{\arabic{exerci}.} \newenvironment{Ex}{\begin{list}% {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt% \labelwidth=12pt\labelsep=6pt\itemsep=2.5ex}}{\end{list}\medbreak} \newenvironment{parts}{\begin{enumerate}\vspace{1ex}\renewcommand{\itemsep}{1ex} \renewcommand{\labelenumi}{(\alph{enumi})}}{\end{enumerate}} \newcommand{\hint}{\textbf{Hint:} } \begin{document} \thispagestyle{empty} %%%%%%% TITLE %%%%%% \begin{center}{\bf PM822 \qquad Assignment 3 \qquad Due Tuesday, February 26}\end{center} \vskip 1em \begin{Ex} \item Let $\fA$ be a non-unital C*-algebra, and let $(e_\lambda)_{\lambda \in \Lambda}$ be an approximate identity. \begin{parts} \item Let $f$ be a state on $\fA$, and let $\pi_f$ be the GNS representation. Show that the vectors $(\dot e_\lambda)$ form a Cauchy net, and that the limit vector $\xi$ is cyclic for $\pi_f(\fA)$. \item Show that if $0 \le a \in \fA$ and $\|a\|=1$, then there is a state $f$ on $\fA$ such that $f(a) = 1$. \item Deduce (as in the unital case) that $\fA$ has a faithful $*$-representation. \end{parts} \item \begin{parts} \item Let $\fA$ be a unital C*-algebra, and let $a \in \fA$ with $\|a\| \le 1$. Prove that the map from $\bC[z] + \bC[\bar z]$ into $\fA$ given by $\psi\big( p(z) + q(\bar z)\big) = p(a) + q(a^*)$ extends to a unital completely positive map of $\rC(\bT)$ into $\fA$. \item Show that if $\phi$ is a positive unital map of $\fA$ into $\B(\H)$, then $\|\phi\| = 1$.\\ \hint consider $\phi \circ \psi$ where $\psi$ comes from (a). \end{parts} \item Let $\fA, \fB$ be C*-algebras, and let $\phi:\fA \to \fB$ be a unital completely positive map. Let $(\pi,V)$ be the Stinespring dilation of $\phi$ \begin{parts} \item Show that if $a \in \fA$ and $\phi(a^*)\phi(a) = \phi(a^*a)$, then $V\H$ is invariant for $\pi(a)$. Hence show that $\phi(ba) = \phi(b)\phi(a)$ for all $b \in \fA$. \item Show that $\fC = \{c \in \fA : \phi(c^*)\phi(c) = \phi(c^*c) \AND \phi(c)\phi(c^*) = \phi(cc^*) \}$ is a unital C*-algebra, and $\phi(c_1ac_2) = \phi(c_1)\phi(a)\phi(c_2)$ for all $a \in \fA$ and $c_1, c_2 \in \fC .$ \end{parts} \item Let $\bA = \{ z \in \bC : r_1 \le |z| \le r_2 \}$ where $0 < r_1 < 1 < r_2 < \infty$. Let $\A = R(\bA)$. Define a functional on $\A$ by \[ \phi(f) = \frac1{2\pi} \int_0^{2\pi} f(e^{it}) \,dt .\] \begin{parts} \item Show that there is more than one state on $\rC(\partial \bA)$ extending $\phi$. \item Show that $\A$ is not Dirichlet. \item Show that $\phi$ has two inequivalent Stinespring dilations. \end{parts} \item A von Neumann algebra is a unital C*-subalgebra $\fN$ of $\B(\H)$ which is weak-$*$-closed. Say that $\fN$ is \textit{semidiscrete} if there is a net of unital weak-$*$ continuous completely positive maps $\phi_\lambda : \fN \to \M_{k_\lambda}$ and $\psi_\lambda : M_{k_\lambda} \to \fN$ such that \[ a = \wslim_\lambda \psi_\lambda( \phi_\lambda(a) ) \qforal a \in \fN .\] Prove that $\fN$ is injective; i.e.\ if $\phi:\S \to \fN$ is a completely positive map of an operator system $\S$ into $\fN$ and $\T$ is an operator system containing $\S$, then there is a completely positive map of $\T$ into $\fN$ extending $\phi$. \end{Ex} \end{document}