\documentclass[12pt]{amsart}
\usepackage{amssymb, amstext, amscd, amsmath}
\pagestyle{empty}
\topmargin 7mm
\textheight 9.7in
\textwidth 6in
\oddsidemargin 0.25in
\evensidemargin 0.25in
\voffset -15mm

%      Blackboard bold letters
\newcommand{\bA}{{\mathbb{A}}}
\newcommand{\bC}{{\mathbb{C}}}
\newcommand{\bD}{{\mathbb{D}}}
\newcommand{\bN}{{\mathbb{N}}}
\newcommand{\bQ}{{\mathbb{Q}}}
\newcommand{\bR}{{\mathbb{R}}}
\newcommand{\bT}{{\mathbb{T}}}
\newcommand{\bZ}{{\mathbb{Z}}}

%      Capital script letters
  \newcommand{\A}{{\mathcal{A}}}
  \newcommand{\B}{{\mathcal{B}}}
  \newcommand{\C}{{\mathcal{C}}}
  \newcommand{\D}{{\mathcal{D}}}
  \newcommand{\E}{{\mathcal{E}}}
  \newcommand{\F}{{\mathcal{F}}}
  \newcommand{\G}{{\mathcal{G}}}
\renewcommand{\H}{{\mathcal{H}}}
  \newcommand{\I}{{\mathcal{I}}}
  \newcommand{\J}{{\mathcal{J}}}
  \newcommand{\K}{{\mathcal{K}}}  
\renewcommand{\L}{{\mathcal{L}}}
  \newcommand{\M}{{\mathcal{M}}}
  \newcommand{\N}{{\mathcal{N}}}
\renewcommand{\O}{{\mathcal{O}}}
\renewcommand{\P}{{\mathcal{P}}}
  \newcommand{\Q}{{\mathcal{Q}}}
  \newcommand{\R}{{\mathcal{R}}}
\renewcommand{\S}{{\mathcal{S}}}
  \newcommand{\T}{{\mathcal{T}}}
  \newcommand{\U}{{\mathcal{U}}}
  \newcommand{\V}{{\mathcal{V}}}
  \newcommand{\W}{{\mathcal{W}}}
  \newcommand{\X}{{\mathcal{X}}}
  \newcommand{\Y}{{\mathcal{Y}}}
  \newcommand{\Z}{{\mathcal{Z}}}

%Greek Letters
\newcommand{\ep}{\varepsilon}
\renewcommand{\phi}{\varphi}
\newcommand{\upchi}{{\raise.35ex\hbox{$\chi$}}}

% Fraktur letters
\newcommand{\fA}{{\mathfrak{A}}}
\newcommand{\fB}{{\mathfrak{B}}}
\newcommand{\fC}{{\mathfrak{C}}}
\newcommand{\fD}{{\mathfrak{D}}}
\newcommand{\fE}{{\mathfrak{E}}}
\newcommand{\fF}{{\mathfrak{F}}}
\newcommand{\fG}{{\mathfrak{G}}}
\newcommand{\fH}{{\mathfrak{H}}}
\newcommand{\fI}{{\mathfrak{I}}}
\newcommand{\fJ}{{\mathfrak{J}}}
\newcommand{\fK}{{\mathfrak{K}}}
\newcommand{\fL}{{\mathfrak{L}}}
\newcommand{\fM}{{\mathfrak{M}}}
\newcommand{\fN}{{\mathfrak{N}}}
\newcommand{\fO}{{\mathfrak{O}}}
\newcommand{\fP}{{\mathfrak{P}}}
\newcommand{\fQ}{{\mathfrak{Q}}}
\newcommand{\fR}{{\mathfrak{R}}}
\newcommand{\fS}{{\mathfrak{S}}}
\newcommand{\fT}{{\mathfrak{T}}}
\newcommand{\fU}{{\mathfrak{U}}}
\newcommand{\fV}{{\mathfrak{V}}}
\newcommand{\fW}{{\mathfrak{W}}}
\newcommand{\fX}{{\mathfrak{X}}}
\newcommand{\fY}{{\mathfrak{Y}}}
\newcommand{\fZ}{{\mathfrak{Z}}}

\newcommand{\FOR}{\text{ for }}
\newcommand{\FORAL}{\text{ for all }}
\newcommand{\AND}{\text{ and }}
\newcommand{\IF}{\text{ if }}
\newcommand{\OR}{\text{ or }}
\newcommand{\qand}{\quad\text{and}\quad}
\newcommand{\qor}{\quad\text{or}\quad}
\newcommand{\qfor}{\quad\text{for}\quad}
\newcommand{\qforal}{\quad\text{for all}\quad}
\newcommand{\qif}{\quad\text{if}\quad}

%Operators
\newcommand{\ad}{\operatorname{ad}}
\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\Alg}{\operatorname{Alg}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\conv}{\operatorname{conv}}
\newcommand{\cconv}{\ol{\operatorname{conv}}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\Dim}{\operatorname{dim}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Homeo}{\operatorname{Homeo}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\Id}{{\operatorname{Id}}}
\newcommand{\Lat}{\operatorname{Lat}}
\newcommand{\nul}{\operatorname{null}}
\newcommand{\OA}{\operatorname{OA}}
\newcommand{\ran}{\operatorname{Ran}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\re}{\operatorname{Re}}
\newcommand{\Rep}{\operatorname{Rep}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\spn}{\operatorname{span}}
\newcommand{\spr}{\operatorname{spr}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\sumoplus}{\operatornamewithlimits{\sum\oplus}}
\newcommand{\Tr}{\operatorname{Tr}}

%      Useful shortforms
\newcommand{\bsl}{\setminus}
\newcommand{\bv}{\bigvee}
\newcommand{\bw}{\bigwedge}
\newcommand{\ca}{\mathrm{C}^*}
\newcommand{\rC}{\mathrm{C}}
\newcommand{\dcup}{\dot{\cup}}
\newcommand{\dint}{\displaystyle\int}
\newcommand{\dlim}{\displaystyle\lim\limits}
\newcommand{\dsum}{\displaystyle\sum\limits}
\newcommand{\dprod}{\displaystyle\prod\limits}
\newcommand{\lip}{\langle}
\newcommand{\rip}{\rangle}
\newcommand{\ip}[1]{\lip #1 \rip}
\newcommand{\bip}[1]{\big\lip #1 \big\rip}
\newcommand{\Bip}[1]{\Big\lip #1 \Big\rip}
\newcommand{\norm}[1]{\left\| #1 \right\|}
\newcommand{\ol}{\overline}
\newenvironment{sbmatrix}{\left[\begin{smallmatrix}}{\end{smallmatrix}\right]}
\newenvironment{spmatrix}{\left(\begin{smallmatrix}}{\end{smallmatrix}\right)}
\newcommand{\td}{\widetilde}
\newcommand{\wh}{\widehat}
\newcommand{\wlim}{\textrm{w--\!}\lim}
\newcommand{\wslim}{\textrm{w*\!--\!}\lim}
\newcommand{\sot}{\textsc{sot}}
\newcommand{\wot}{\textsc{wot}}
\newcommand{\wotclos}[1]{\ol{#1}^{\textsc{wot}}}
\newcommand{\sotclos}[1]{\ol{#1}^{\textsc{sot}}}
\newcommand{\wsclos}[1]{\ol{#1}^{\text{weak-}*}}
\DeclareMathOperator*{\wotlim}{\textsc{wot}--lim}
\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}