0$ such that $\|f_n\|\leq K$ for all $n$. \end{itemize} \section{Basics of Banach algebras; constructions; group of units} \begin{itemize} \item Let $A$ be an algebra. Suppose that there are $a\in A$ and a sequence $(b_n)$ in $A$, each $b_n$ is non-zero, such that $ab_n = nb_n$ for all $n$. Show that there is no algebra norm on $A$. Use this result to show that $C(\mathbb R)$, the algebra of all continuous functions on $\mathbb R$, cannot be given an algebra norm. \item Let $A$ be a commutative Banach algebra such that for each $a\in A$, there is $n\in\mathbb N$ with $a^n=0$. Prove that there is $N\in\mathbb N$ with $a^N=0$ for all $a\in A$. \emph{Hint:} Baire Category. Can you prove the same for a non-commutative Banach algebra? \end{itemize} \section{Spectrum; Characters; Gelfand Theory} \begin{itemize} \item Let $A$ be a Banach algebra, and let $a,b\in A$. Show that $\Sp(ab)\setminus\{0\} = \Sp(ba)\setminus\{0\}$ (this is probably in the book-- check that you understand the proof!) Can it happen that $\Sp(ab)\not=\Sp(ba)$? Give a proof (by contradiction!) that $ab-ba$ cannot be a multiple of $1$ (assuming that $A$ is unital). \item Find examples of $2\times 2$ complex matrices $A,B$ such that $\rho(AB) > \rho(A)\rho(B)$ and $\rho(A+B) > \rho(A) + \rho(B)$. \emph{Hint:} Remember that $\Sp(A)$ is just the collection of eigenvalues of $A$. \item Let $A$ be a Banach algebra, and suppose that for $C>0$, we have that $\|a\| \leq C \rho(a)$ for all $a\in A$. Show that $A$ is commutative. \emph{Hint:} Let $a,b\in A$, and define $f(z) = e^{-za} b e^{za}$, for $z\in\mathbb C$. Prove that $f$ is analytic and constant. Deduce the result from this. \end{itemize} \section{Commutative Banach algebras; holomorphic functional calculus} \begin{itemize} \item Let $A$ be a Banach algebra, let $a\in A$, and suppose that $0$ and $\infty$ belongs to the same unbounded component of $\mathbb C\setminus\Sp(a)$. Show that: \begin{enumerate} \item $a=e^b$ for some $b\in A$; \item for any $n\in\mathbb N$ there is $c\in A$ with $c^n = a$. \item for $\epsilon>0$, we can find a complex polynomial $P$ such that $\| a^{-1} - P(a) \| < \epsilon$. \end{enumerate} Show that if $M$ is an $n\times n$ invertible matrix, then $M=e^L$ for some matrix $L$. \end{itemize} \section{C$^*$-algebras; continuous functional calculus} \begin{enumerate} \item Let $A$ be a C$^*$-algebra, and let $a\in A$. Supposing that $a$ is normal, show that $\Sp(a^*a) = \{ |\lambda|^2 : \lambda\in\Sp(a) \}$. Is this always true if $a$ is not normal? \item Let $X$ be a compact Hausdorff space, let $A=C(X)$ with the usual norm. Let $\|\cdot\|_0$ be some other algebra norm on $A$ (we do not assume that $(A,\|\cdot\|_0)$ is Banach). Show that: \begin{enumerate} \item Let $B$ be the completion of $(A,\|\cdot\|_0)$, so that $B$ is a Banach algebra. Let $E$ be the collection of all characters $\varphi$ on $B$, restricted to the algebra $A$. Show that $E$ forms a non-empty, closed subset of the character space of $A$ (which we identify with $X$). \item Using Urysohn's Lemma, show that if $E\not=X$, then there are non-zero $a,b\in A$ with $ab=0$ but with $\varphi(a)=1$ for all $\varphi\in E$. Show that this leads to a contradiction; so $E=X$. \item Deduce that for each $f\in A$, we have $\|f\| = \rho_B(f)$. \item Deduce that $\|f\| \leq \|f\|_0$ for each $f\in A$. \end{enumerate} \item Let $X,Y$ be compact Hausdorff spaces, and let $T:C(X)\rightarrow C(Y)$ be a unital homomorphism. Show that there is a continuous map $f:Y\rightarrow X$ such that $T(a) = a\circ f$ for all $a\in C(X)$. If you know what the words mean: Show that the category of compact Hausdorff spaces with continuous maps is anti-equivalent to the category of unital commutative C$^*$-algebras with unital homomorphisms. \item In the book, Corollary~2.19 is stated for C$^*$-algebras $A$ and $B$. Prove that the result still holds if $A$ is merely a Banach $*$-algebra. \item Consider the Hilbert space $H = \ell^2 = \ell^2(\mathbb N)$, with the standard orthonormal basis $(e_n)$ (so $e_1=(1,0,0,\cdots), e_2=(0,1,0,\cdots)$ and so forth). Let $(a_n)$ be a sequence of complex numbers. Show that there is a bounded linear operator $T$ on $H$ with $T(e_n) = a_n e_n$ for all $n$, if and only if $(a_n)$ is a bounded sequence. Show that $T$ is a normal operator. In terms of the sequence $(a_n)$, determine when $T$ is: (i) unitary, (ii) self-adjoint. \item We continue with the same notation. For $T$ defined by a sequence $(a_n)$, determine the spectrum of $T$. \item We continue with the same notation. Let $A$ be the C$^*$-algebra (in $\mathcal B(H)$) generated by $T$. Show that: \begin{enumerate} \item As $T^*T = TT^*$, we can talk about a ``polynomial in $T$ and $T^*$''. Show that the collection of all such polynomials, $\mathbb C[T,T^*]$ is dense in $A$. \emph{Hint:} By definition, $A$ is the smallest C$^*$-algebra containing $T$. Show that any C$^*$-algebra containing $T$ contains $\mathbb C[T,T^*]$, and then show that the closure of $\mathbb C[T,T^*]$ is a C$^*$-algebra. \item It follows that $A$ is commutative. Using the results of Section~6.4 in the book, show that if $\varphi\in\Phi_A$, then $\varphi$ is uniquely determined by the value $\varphi(T)$. \item By Commutative Gel'fand--Naimark (Theorem~6.24) $A$ is isomorphic to $C(\Phi_A)$. Show that the compact Hausdorff spaces $\Phi_A$ and $\Sp(T)$ are homeomorphic. \emph{Hint:} Show firstly that the map $\Phi_A\rightarrow \Sp(T); \varphi\mapsto\varphi(T)$ is well-defined and injective. Now prove that it is surjective (and then appeal to the result that a continuous bijection between compact, Hausdorff spaces is a homeomorphism). \end{enumerate} \item We continue with the same notation. Let $f$ be a continuous function on the spectrum of $T$, so by the Continuous Functional Calculus, we can make sense of $f(T)$. Now consider the map $\Phi:C(\Sp(T)) \rightarrow \mathcal B(H)$ which maps $f$ to $S$, where \[ S(e_n) = f(a_n) e_n \qquad \text{for all $n$}. \] Using the previous two questions, show that this is well-defined (that is, $f(a_n)$ makes sense, and that $S$ is bounded). Show that $\Phi$ is a unital $*$-homomorphism with $\Phi(Z)=T$. Conclude that $\Phi$ agrees with the Continuous Functional Calculus. \emph{Remark:} So in this case, we have a very concrete picture of what the Continuous Functional Calculus actually is! \end{enumerate} \section{Representation theory; modules; radicals; uniqueness of norm} \begin{itemize} \item I find the discussion in Section~5.3 hard to follow. Check \emph{carefully} that you understand why the definition of the Radical given for commutative algebras on page~193 agrees with the general definition give on page~232. \item This one is in the book, but let's try to give a nicer proof. Firstly, check that you understand that a unital commutative Banach algebra $A$ is semisimple if and only if the Gelfand transform $\mc G:A\rightarrow C(\Phi_A)$ is injective. \smallskip \noindent\textbf{Theorem:} Let $A$ and $B$ be unital commutative Banach algebras, with $B$ semisimple. Then any unital homomorphism $T:A\rightarrow B$ is continuous. \smallskip Here is a strategy for proving this: \begin{itemize} \item Let $\varphi$ be a character on $B$. Show that $\phi = \varphi\circ T$ is a character on $A$, and hence conclude that $\phi$ is bounded. \item Let $(a_n)$ be a sequence in $A$ converging to $0$, and suppose that $b = \lim_n T(a_n)$ exists in $B$. Show that $\mc G(b)=0$, and hence that $b=0$. \item Use the closed graph theorem to conclude that $T$ is continuous. \item Now write all that up neatly! \end{itemize} \item Check that you understand why this result implies that a unital commutative semisimple Banach algebra has a unique (complete algebra) norm. \end{itemize} \section{Applications and examples to group algebras} \section{More additional questions on later parts of the course} \begin{itemize} \item Let $u$ be a unitary element in a unital C$^*$-algebra $A$. Suppose that $\Sp(u)$ is not the whole of the unit circle. Show that there is $a\in A$ with $a^*=a$ and $u = \exp(ia)$. \emph{Hint:} Functional calculus. \item Let $\mathbb T$ be the unit circle in $\mathbb C$, and let $u\in C(\mathbb T)$ be the element $u(z)=z$. Show that there is no $a\in C(\mathbb T)$ with $u=\exp(ia)$. \item Let $T,S\in\mc B(H)$ satisfy $T^*T \leq S^*S$. (Recall that for $A,B\in\mc B(H)$ we define $A\leq B$ to mean that $(Ax|x) \leq (Bx|x)$ for all $x\in H$). Show that there exists $U\in\mc B(H)$ with $T=US$ and $\|U\|\leq 1$. \emph{Hint:} Show that $U:S(H)\rightarrow H; S(x)\mapsto T(x)$ is well-defined, linear, and bounded. Extend $U$ to all of $H$ by orthogonal decomposition. \end{itemize} \end{document}