% LaTeX file -- PM 810, Winter 2019, Assignment 2

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 810 \hfill Assignment 2 \hfill
Due Thursday February 14.}} 
\vskip 2em 

\begin{Ex}

\item Let $\ell_p$, for $1 \le p < \infty$, be a Banach algebra with pointwise multiplication.
\begin{parts}
\item Find its maximal ideal space.
\item Prove that it has maximal ideals which are not modular. \hint $(l_p)^2$ is a proper ideal.
\end{parts}

\item Let $X$ be a compact Hausdorff space, and let $\I$ be an ideal of $\rC(X)$, not necessarily closed.
Let $E = \ker \I = \{x\in X : f(x)=0 \FORAL f\in\I\}$.  
Let $I_0(E)$ be the ideal of functions that
vanish on an open neighbourhood of $E$, 
and let $I(E)$ denote the ideal of functions which vanish on $E$.
Prove that $I_0(E) \subset \I \subset I(E)$.

\item Let $\fA$ and $\fB$ be unital commutative Banach algebras.
\begin{parts}
\item Suppose that $\theta: \fA \to \fB$ is an injective homomorphism, and $\fB$ is semisimple.
Prove that $\fA$ is semisimple. \hint if $a \in \rad(\fA)$, what can you say about $\sigma(\theta(a))$?

\item Let $\fA = C^1[0,1]$ with $\|f\|=\|f\|_\infty + \|f'\|_\infty$. Let $\J = \{f\in\fA: f(0)=f'(0)=0 \}$.
Show that $\fA$ is semisimple but $\fA/\J$ is not.
\end{parts}

\item Let $\fA$ be a unital commutative Banach algebra.  If $a_1,\dots,a_n\in\fA$,
define the \textit{joint spectrum} to be 
$
 \sigma(a_1,\dots,a_n) = \{ \lambda = (\phi(a_1),\dots,\phi(a_n)) \in \bC^n : \phi \in \M(\fA) \} .
$
\begin{parts}
\item Show that $\lambda = (\lambda_1,\dots,\lambda_n)\not \in  \sigma(a_1,\dots,a_n)$
if and only if there are $b_1,\dots, b_n\in\fA$ such that $\sum_{i=1}^n (a_i-\lambda_i)b_i = 1$.

\item We say $a_1,\dots,a_n$ generate $\fA$ if the set of polynomials in $a_1,\dots,a_n$ is dense in $\fA$.  If this occurs, show that $\Phi : \M(\fA) \to \sigma(a_1,\dots,a_n)$
given by $\Phi(\phi) = (\phi(a_1),\dots,\phi(a_n))$ is a homeomorphism.

\item If $a_1,\dots,a_n$ generate $\fA$, show that $ \sigma(a_1,\dots,a_n)$ is \textit{polynomially convex}; i.e. if $\lambda\in\bC^n$ and 
$|p(\lambda)| \le \sup\{ |p(z)| : z \in \sigma(a_1,\dots,a_n) \}$, 
then $\lambda \in \sigma(a_1,\dots,a_n)$.
\hint if $\lambda \not \in  \sigma(a_1,\dots,a_n)$, use part (a) and
approximate $1 - \sum_{i=1}^n (a_i-\lambda_i)b_i = 0$ by a polynomial.

\end{parts}

%\item Let $\fA = l_2 \oplus \bC r$ with $\|(a_n)+\alpha r\| = \|(a_n)\|_2 + |\alpha|$.  
%Define $\big((a_n)+\alpha r\big) \big((b_n)+\beta r\big) = (a_nb_n)$. %\vspace{-2ex}
%\begin{parts}
%\item Find $\rad\fA$.
%\item Show that $\fA^2 = l_1$.  
%\item Pick any non-zero linear functional $\psi$ on $l_2$ which is zero on $l_1$.  
%Define a new norm $|\!|\!|(a_n)+\alpha r|\!|\!| = \|(a_n)\|_2 + |\psi(a_n)-\alpha|$.
%Show that $(\fA,|\!|\!|\cdot|\!|\!|)$ is complete, making $|\!|\!|\cdot|\!|\!|$ a Banach algebra
%norm on $\fA$ which is not equivalent to the original.
%\end{parts}

\item Let $\fA$ be a (non-commutative) Banach algebra.
\begin{parts}
\item Prove that $\fA/\rad(\fA)$ is semisimple.
\item If $\fJ$ is a closed ideal of $\fA$, show that $\rad(\fJ) = \fJ \cap \rad(\fA)$.
\end{parts}

\item Let $\fA, \fB$ be (non-commutative) Banach algebras, and let $\theta:\fA\to\fB$
be a surjective homomorphism.
\begin{parts}
\item Show that $\theta(\rad(\fA)) \subset \rad(\fB)$.
\item If $\fB$ is semisimple, prove that $\ker\theta$ is closed.
\end{parts}

\end{Ex}
\end{document}
 

 