\documentclass[12pt]{amsart}
\usepackage{amssymb, amstext, amscd, amsmath}
\pagestyle{empty}
\topmargin 1cm
\textheight 9.2in
\textwidth 6in
\oddsidemargin .25in
\evensidemargin .25in
\voffset -1.5cm

%      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 1 
\qquad Due Tuesday, January 22}\end{center}  
\vskip 2em
 
\begin{Ex}
\item  Let $\bA$ denote the set of all continuous functions with
absolutely convergent Fourier series with the norm $\|f\|_\bA := \sum_{n \in \bZ} |\hat f(n)|$.
\begin{parts}
\item Prove that $\bA(\bT)$ is a Banach algebra which is isometrically isomorphic to $\ell_1(\bZ)$ with convolution.
\item Identify the maximal ideal space of $\bA$.\\
\hint for $\phi\in\M_{\bA}$, show that $\lambda = \phi(z)$ determines $\phi$.
\item Prove Wiener's Theorem: if $f \in \bA$ and $f(z) \ne 0$ for all $|z|=1$, then $1/f$ also has
an absolutely convergent Fourier series.
\end{parts}


\item Let $\X$ be a Banach space. Let $\Omega$ be an open subset of $\bC$.
Let $f:\Omega \to \X$ be a function which is \textit{weakly analytic}, 
meaning that $\phi\circ f$ is analytic for every continuous linear functional $\phi\in\X'$.
Fix $z_0\in\Omega$ and a closed disk $D=\ol{b_r(z_0)} \subset \Omega$.
\begin{parts}
\item Prove that $f$ is bounded on $D$. \hint uniform boundedness principle.
\item Use Cauchy's Theorem to express $\dfrac{\phi(f(z_0+h)) - \phi(f(z_0))}{h}- (\phi\circ f)'(z_0)$
as an integral around the boundary of $D$ for all $|h|<r$. Hence show that for $\|\phi\|\le1$,
this difference tends to $0$ uniformly as $|h|\to0$.
\item Deduce that $\dfrac{f(z_0+h) - f(z_0)}{h}$ is Cauchy as $h\to0$, and hence that $f$ is differentiable (i.e. analytic).
In particular, $f$ is continuous and therefore Riemann integrable.
\item Show that Cauchy's Theorem is valid for $\X$-valued analytic functions: if $\C$
is a rectifiable contour in $\Omega$ which is homologous to zero, then
$\int_\C f(z)\,dz = 0$. \\[.6ex]
\textit{Remarks: $\bullet$\ This is a Riemann integral, which makes sense for continuous functions
into $\X$ just as for scalar functions.\ 
$\bullet$\ All standard results about analytic functions now follow using the classical proofs.}
\end{parts}


\item 
\begin{parts}
\item (Fuglede) Show that if $N \in \B(\H)$ is normal, and commutes with $X \in \B(\H)$,
then $N^*$ also commutes with $X$. 
\hint consider the entire analytic function \\ \strut\qquad\qquad \qquad
$f(z) = e^{zN^*} X e^{-zN^*} = e^{zN^*} e^{-\bar{z}N}  X e^{\bar{z}N} e^{-zN^*} $.
\item Deduce that if $\{N_i: i \in \I\}$ is a family of commuting normal operators, then
$\fA = \ca(\{N_i : i \in \I\})$ is abelian. Show that the maximal ideal space of $\fA$ can be
naturally identified with a compact subset $X$ of $\prod_{i\in\I} \sigma(N_i)$ such that 
the projection $\pi_i(X)$ onto the $i$-th coordinate is surjective for each $i\in\I$.
\item If $U_1,\dots,U_n$ are commuting unitary operators, show that there is a $*$-representation
of $\rC(\bT^n)$ given by $\rho(f) = f(U_1,\dots,U_n)$. Hence deduce that for any polynomial
$p\in\bC[z_1,\dots,z_n]$, we have $\|p(U_1,\dots,U_n)\| \le \|p\|_\infty := \sup_{|z_i|=1, 1 \le i \le n} |p(z_1,\dots,z_n)|$.
\end{parts}




\end{Ex}

\end{document}