% LaTeX file -- PM 810, Winter 2014, Assignment 4
\documentclass[12pt]{amsart}
\usepackage{amssymb, amstext, amscd, amsmath}
\pagestyle{empty}
\topmargin 2cm
\textheight 9.0in
\textwidth 6.0in
\oddsidemargin 0.25in
\evensidemargin 0.25in
\voffset -3.0cm

%      Blackboard bold letters
\newcommand{\bA}{{\mathbb{A}}}
\newcommand{\bB}{{\mathbb{B}}}
\newcommand{\bC}{{\mathbb{C}}}
\newcommand{\bD}{{\mathbb{D}}}
\newcommand{\bE}{{\mathbb{E}}}
\newcommand{\bF}{{\mathbb{F}}}
\newcommand{\bG}{{\mathbb{G}}}
\newcommand{\bH}{{\mathbb{H}}}
\newcommand{\bI}{{\mathbb{I}}}
\newcommand{\bJ}{{\mathbb{J}}}
\newcommand{\bK}{{\mathbb{K}}}
\newcommand{\bL}{{\mathbb{L}}}
\newcommand{\bM}{{\mathbb{M}}}
\newcommand{\bN}{{\mathbb{N}}}
\newcommand{\bO}{{\mathbb{O}}}
\newcommand{\bP}{{\mathbb{P}}}
\newcommand{\bQ}{{\mathbb{Q}}}
\newcommand{\bR}{{\mathbb{R}}}
\newcommand{\bS}{{\mathbb{S}}}
\newcommand{\bT}{{\mathbb{T}}}
\newcommand{\bU}{{\mathbb{U}}}
\newcommand{\bV}{{\mathbb{V}}}
\newcommand{\bW}{{\mathbb{W}}}
\newcommand{\bX}{{\mathbb{X}}}
\newcommand{\bY}{{\mathbb{Y}}}
\newcommand{\bZ}{{\mathbb{Z}}}

\newcommand{\bk}{{\mathbf{k}}}
\newcommand{\rC}{{\mathrm{C}}}

%      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}}}

%Gothic 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}

\renewcommand{\ae}{\operatorname{a.e.}}
\newcommand{\Ch}{\operatorname{Ch}}
\newcommand{\Dim}{\operatorname{dim}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Ext}{\operatorname{Ext}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\ind}{\operatorname{ind}}
\newcommand{\ol}{\overline}
\newcommand{\rad}{\operatorname{rad}}
\newcommand{\ran}{\operatorname{Ran}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\spn}{\operatorname{span}}
\newcommand{\spr}{\operatorname{spr}}
\newcommand{\sot}{\textsc{sot}}
\newcommand{\wot}{\textsc{wot}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\ep}{\varepsilon}

\newcommand{\ca}{\mathrm{C}^*}
\newcommand{\CT}{\mathrm{C}(\bT)}
\newcommand{\dint}{\displaystyle\int}
\newcommand{\dlim}{\displaystyle\lim\limits}
\newcommand{\dsum}{\displaystyle\sum\limits}
\renewcommand{\phi}{\varphi}
\newcommand{\upchi}{{\raise.35ex\hbox{$\chi$}}}
\newenvironment{sbmatrix}{\left[\begin{smallmatrix}}{\end{smallmatrix}\right]}

\newcounter{exerci}[section]
\renewcommand{\theexerci}{\arabic{exerci}.}
\newenvironment{Ex}{\begin{list}%
 {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt%
 \labelwidth=12pt\labelsep=6pt\itemsep=3ex}}{\end{list}\medbreak}
\newenvironment{parts}{\vspace{1.5ex}\begin{enumerate}\renewcommand{\itemsep}{.6ex}
 \renewcommand{\labelenumi}{(\alph{enumi})}}{\end{enumerate}}
\newcommand{\hint}{\quad\textbf{Hint: }}

\begin{document}
\thispagestyle{empty}

%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 950 \hfill Assignment 2 \hfill
Due Tuesday October 21}} 
\vskip 2em 

\begin{Ex}

\item 
\begin{parts}
\item If $K$ is a p-set for $A$ and $L\subset K$ is a p-set for $A|_K$, prove $L$ is a p-set for $A$.
\item Show that every p-set with at least 2 points contains a proper subset which is a p-set.
	Hence show that every p-set contains a p-point.
\end{parts}


\item Let $A$ be a function algebra, and let $\Gamma$ be its Shilov boundary.
\begin{parts}
\item  Let $\phi\in\M_A$. Prove that there is a minimal closed subset $E$ of $\Gamma$ supporting a representing measure for $\phi$. 
\item Suppose that $\phi\in \M_A\setminus \Gamma$, and let $E$ be a minimal closed support set found in (a). Prove that $E$ has no isolated points.\\
\hint if $x\in E$ is isolated, pick $f\in A$ with $f(x)=0$ and $\phi(f)=1$.
\end{parts}


\item Let $A_\alpha$ be the algebra from Assignment 1.
Suppose that $B$ is a function algebra so that $A_\alpha \subset B \subset \rC(\bT^2)$.
Show that either $B=A_\alpha$ or $B=\rC(\bT^2)$.

\hint Either $z$ is invertible in $B$ or there is a $\phi\in\M_B$ so that $\phi(z)=0$. 
Look up a proof of Wermer's maximality theorem, and adapt the argument.
See Gamelin, p.38, or Hoffman, \textit{Banach spaces of analytic functions}, p.93.

\item 
\begin{parts}
\item Let $f\in A$ and $\phi\in\M_A$. Show that if $|\phi(f)|=\|f\|$, then $f$ is constant on the Gleason part $P(\phi)$.
\item Let $K$ be a compact subset of a Gleason part $P$. Prove that if $\psi\in\M_A$ satisfies
$|\psi(f)| \le \|f\|_K$ for all $f\in A$, then $\psi\in P$.
\end{parts}


\item Let $\phi\in\M_A$. Suppose that there is a function $g\in\ker\phi$ which has an $n$th root $g_n\in A$ for all $n\ge1$.
Prove that $\hat g$ vanishes on the Gleason part of $\phi$.

\item Let $T:A\to A$ be a surjective isometry.  
\begin{parts} 
\item Show that there is a bijection $\sigma$ of $\Ch(A)$ onto itself and a map $\lambda:\Ch(A) \to \bT$ so that
$ T^*\delta_x = \lambda(x) \delta_{\sigma(x)}$ for all $x \in \Ch(A)$. 
\item Show that $\lambda = T(1)$ restricted to $\Ch(A)$.
\item Show that $\lambda(x) T(fg)(x) = T(f)(x)\,T(g)(x)$ for $x\in\Ch(A)$. Deduce that $\lambda$ is invertible in $A$.
\hint choose $f,g$ to make the right side nice. 
\item Show that $|\lambda(x)|=1$ on all of $\M_A$, and that $T(f) = \lambda \theta(f)$,
where $\theta$ is an isometric automorphism of $A$.
\end{parts}


\end{Ex}
\end{document}
 

 