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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 950 \hfill Assignment 1 \hfill
Due Tuesday Sept. 29.}} 
\vskip 2em 

\begin{Ex}

%\item Suppose that $B$ is a function algebra so that $A(\bD) \subset B \subset \rC(\bT)$.
%Show that either $B=A(\bD)$ or $B=\rC(\bT)$.
%\hint Either $z$ is invertible in $B$, or there is a $\phi\in\M_B$ so that $\phi(z)=0$. In the second case, find
%a representing measure $\mu$  for $\phi$. Since $z^n\mu$ is an annihilating measure for $n\ge1$, show that every $f\in B$ belongs to $A(\bD)$.

\item Let $A$ be a function algebra. Find all extreme points of the unit ball of $A^*$.


\item Prove that if $X$ is metrizable, and $A$ is a function algebra on $X$, then $\Ch(A)$ is the unique minimal boundary set for $A$.

\item Let $\Lambda$ be an uncountable set,  let $X = \prod_{\lambda\in \Lambda} [0,1]$, and let $A = \rC(X)$. 
A point in $X$ may be written $x = (x_\lambda)_{\lambda\in \Lambda} $.
\begin{parts} 
\item Show that every $f \in A$ depends on only countably many coordinates.
\item Show that every point of $X$ is a p-point.
\item Show that $E = \{x \in X : x_\lambda=0 \text{ except countably often}\}$ is a boundary for $A$.
\item Find another boundary which is disjoint from $E$, showing that there is no minimal boundary.
\end{parts}


\item Let $0 < \alpha < 1$ be an irrational number. Consider $\bT^2$ as a subset of $\bC^2$.
Let $A_\alpha$ be the subalgebra of $\rC(\bT^2)$ spanned by $\{ z^mw^n : m,n \in \bZ \AND m + n\alpha \ge 0\}$.

\begin{parts} 
\item Show that $A$ is a Dirichlet algebra, i.e., $\{f + \bar g : f,g \in A\}$ is dense in $\rC(\bT^2)$.

\item Let $\phi \in \M_{A_\alpha}$ and let $r = |\phi(z)|$. Prove that $|\phi(w)| = r^\alpha$.

\item Prove that if $f(z)$ is a bounded analytic function on the right half plane $H_+$ and continuous on $\ol{H_+}$, 
then $\sup_{z\in H_+} |f(z)| = \sup_{y\in \bR} |f(iy)|$.  
\hint Fix a branch $L(z)$  of $\log (z+1)$ on $H_+$.  
For $\ep > 0$, let $g_\ep(z) = f(z) e^{-\ep L(z)}$. This is continuous at $\infty$.

\item If $0 < r \le 1$ and $\theta, \eta \in [0,2\pi)$, show that there is a $\phi\in\M_{A_\alpha}$ such that
$\phi(z) = re^{i\theta}$ and $\phi(w) = r^\alpha e^{i\eta}$. 
\hint if $p(z,w) = \sum a_{m,n} z^m w^n$ is a finite sum of monomials in $A_\alpha$, 
define an analytic function at $u=x+iy \in H_+$ by \\
$f(u)  = \sum a_{m,n}e^{i(m\theta+n\eta)}  e^{-(m + n\alpha) u}
= \sum a_{m,n} e^{-(m+ n\alpha)x}e^{i(\theta-y)m} e^{ i(\eta/\alpha-y)n\alpha}$.\\
Note that $\phi(p) = f(-\log r)$.

\item Decribe $\M_{A_\alpha}$.
\end{parts}


\item Let $\mu$ be a finite measure on $\bC$ with compact support $K$.
Define $ \hat\mu(w) = \int_K \frac{d\mu(z)}{z-w} $.
\begin{parts}
\item Show that $\hat\mu$ is analytic on $\bC\setminus K$ and $\lim_{|w|\to\infty} \hat\mu(w) = 0$.
\item Show that $\hat\mu(w)$ is defined and finite $m_2  \ae$, where $m_2$ is planar Lebesgue measure.
\hint estimate $\int_K |\hat\mu(w)| \,dm_2(w)$.
\item Show that if $X \subset \bC$ is a compact set and $\mu\in M(X)$, 
then $\mu \perp R(X)$ if and only if $\hat\mu = 0$ on $\bC\setminus X$. 
\end{parts}


\item Let $X \subset \bC$ be a compact set. Show that $R(X)$ is doubly generated as a function algebra.
\hint Fix a sequence $a_n$, $n\ge1$, consisting of one point in each bounded component of $\bC\setminus X$.
Choose $\ep_n>0$ decreasing sufficiently rapidly to $0$, and define 
$ f_1(z) = z $ and $f_2(z) = \dsum_{n\ge1} \tfrac{\ep_n}{z-a_n}$. 
Let $B$ be the function algebra generated by $f_1$ and $f_2$.
If $z-a_1$ is not invertible in $B$, take $\phi\in\M_B$ with $\phi(z)=a_1$; and estimate $|\phi((z-a_1)f_2)|$.
If you have defined $(\ep_n)$ properly, you will get a contradiction.

\end{Ex}
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