\documentclass[12pt]{amsart}
\usepackage{amssymb, amstext, amscd, amsmath}
\pagestyle{empty}
\topmargin 7mm
\textheight 9.7in
\textwidth 6.5in
\oddsidemargin 0.0in
\evensidemargin 0.0in
\voffset -25mm

%      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=2ex}}{\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 4 \qquad Due Tuesday, March 11}
\end{center}  
\vskip 1em
 
\begin{Ex}
\item
\begin{parts}
\item  Let $1 \in \M$ be an operator space, let $\phi:\M\to\B(\H)$ be a completely isometric unital map, 
and let $\tilde\phi$ be the completely positive extension to the operator system $\S = \ol{\M+\M^*}$.
Prove that $\tilde\phi$ is completely isometric.
\item Suppose that $\theta$ is a unital complete isometry of operator space $\S_1$ onto $\S_2$.
Show that a unital cp map $\phi:\S_2\to \B(\H)$ has the unique extension property 
if and only if $\phi\theta:\S_1\to\B(\H)$ has the unique extension property.
\end{parts}


\item Let $\N = \big\{ \{0\}=N_0 \subset N_1 \subset \dots \subset N_k = \bC^n \big\}$ 
be a \textit{nest} of subspaces of $\bC^n$.
Let 
\[ \T(\N) = \{ T \in \M_n : TN_i \subset N_i \FOR 1 \le i \le k \} \]
be the \textit{nest algebra} of $\N$.
Let $\rho$ be a contractive unital representation of $\T$.
Note that $\T$ contains a copy of $\T_n$ of all upper triangular matrices.
Show that a $*$-dilation of $\rho|_{\T_n}$ yields a $*$-dilation of $\rho$.
In particular, $\rho$ is completely contractive.


\item Let $1\in\A$ be a unital operator algebra contained in $\fA = \ca(\A)$.
Let $\rho:\A\to\B(\H)$ be a completely contractive representation.
A coextension $\sigma$ of $\rho$ into $\B(\K)$ is \textit{extremal} if every 
coextension of $\sigma$ has the form $\tau=\sigma\oplus\tau'$ on $\L=\K \oplus \L'$. 
Extremal extensions are defined analogously.
\begin{parts}
\item Prove that $\rho$ has an extremal coextension.\\
\textbf{Bonus.}  If you are careful, you can also arrange that
$\H$ is \textit{cyclic}: $\K=\ol{\sigma(\A)\H}$.
\item Show that a dilation of $\rho$ is a maximal dilation if and only if it is
an extremal coextension and an extremal extension.
\item Starting with $\rho$, find an extremal coextension $\sigma_1$, 
then find an extremal extension $\tau_1$ of $\sigma_1$. Repeat this process recursively.
Show that the limiting process yields a maximal dilation $\pi$.
\item Show that if $\rho$ is an extremal coextension, and $\pi$ is a maximal dilation of $\rho$, 
then $\H$ is an invariant subspace for $\pi(\A)$.\\ 
(\textit{Note: the converse is not true in general. It is sometimes true---see} 5(b).)
\end{parts}


\item Let $\{e_n : n\in\bZ\}$ be an orthonormal basis for $\H$, and let $E_{ij}$ be the
corresponding matrix units. Let $\A = \ol{\spn\{I, E_{ii}, E_{2i+1,2i} , E_{2i-1,2i}: i \in\bZ\}}$.
\begin{parts}
\item 
Let $\rho(E_{00}) = \rho(I) = 1$ and $\rho(E_{ij})=0$ otherwise be a representation of $\A$ into $\bC$.
Find all irreducible extremal coextensions and extremal extensions of $\rho$.
\item Show that the process of 3(c) takes countably many steps using cyclic (co)-extensions,
but can also be done by one extremal coextension and one extremal extension.
\item What is the C*-envelope of $\A$? Find all boundary representations of $\A$.
\end{parts}


\item Suppose that an operator algebra $\A \subset \ca_e(\A)$ is \textit{semi-Dirichlet},
meaning  $\A^*\A \subset \ol{\A+\A^*}$.
\begin{parts}
\item Let $\rho$ be a representation of $\A$, let $\pi_i$ be two maximal $*$-dilations of $\rho$ on $\L_i$,
let $\K_i = \ol{\pi_i(\A)\H}$, and let $\sigma_i(a) = P_{\K_i}\pi(a)|_{\K_i}$. 
Prove that these two representations are unitarily equivalent via a unitary that fixes $\H$.

\item \textbf{Bonus.} Use problem 3 to show that the representation $\sigma_1$ is extremal.

\item Show that if $\A$ and $\A^*$ are both semi-Dirichlet, then $\A$ is Dirichlet
(meaning that $\ol{\A+\A^*} = \ca_e(\A)$).
\end{parts}

\end{Ex}

\end{document}