% LaTeX file -- PM 810, Winter 2019, Assignment 3

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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 810 \hfill Assignment 3 \hfill
Due Thursday March 14.}} 
\vskip 2em 

\begin{Ex}

\item 
\begin{parts} 
\item Show that if $\A$ is a separable C*-algebra, then it has an approximate unit consisting of an increasing sequence.

\item Show that if $0 \le a$, $0 \le b$ and $\|a\|<1$ and $\|b\|<1$, then \\[1ex]
\strut\hfill $c = b + (1-b)^{1/2} [(1-b)^{-1/2}(a-b)(1-b)^{-1/2}]_+ (1-b)^{1/2}$  \hfill\strut \\[1ex]
satisfies $b\le c$, $a \le c$ and $\|c\|<1$.
\end{parts}

\item Show that if $\J$ is a closed ideal of a C*-algebra $\A$ and
$a\in \A$, then there is an element $j\in \J$ such that
$\|a - j\|=\dist(a,\J)$.  
\hint  write $|a|-\|a+\J\|1$ as $b_+-b_-$.

\item 
Show that if $S$ and $T$ are two normal operators, then there is a
$*$-isomorphism between $\ca(S)$ and $\ca(T)$ taking $S$ to $T$ if and
only if $\sigma(S)=\sigma(T)$.

\item 
\begin{parts}
\item The ultraweak (or $\sigma$-weak) topology is the weakest topology on $\B(H)$ such that the functionals $\omega(T) = \sum_{n\ge1} \ip{Tx_n,y_n}$ are continuous for all families $\{x_n,y_n : n\ge1\}$ of vectors such that $\sum_{n\ge1} \|x_n\|\,\|y_n\| < \infty$. Likewise the ultrastrong topology is the  topology determined by  seminorms of the form
$\rho(T) = \big(\sum_{n\ge1} \|Tx_n\|^2 \big)^{1/2}$ where $\sum_{n\ge1} \|x_n\|^2 < \infty$. Prove that these two topologies have the same continuous linear functionals.

\item Find a linear functional on $\B(H)$ which is ultraweak continuous but not WOT continuous. Hence find a subspace of $\B(H)$ such that its ultraweak closure is properly contained in its WOT-dense closure.

\item Rework the proof of von Neumann's Double Commutant Theorem to show that if $\A$ is a non-degenerate C*-subalgebra of $\B(H)$, then $\A''$ equals the ultraweak closure of $\A$.
\end{parts}

\item Let $\A \subset\B(H)$ be a concrete C*-algebra.
Suppose that $A\in\A$ has a polar decomposition $A=U|A|$.
Show that $Uf(|A|)$ belongs to $\A$ provided that $f \in \rC(\sigma(|A|))$
and $f(0)=0$.

\item 
\begin{parts}
\item If $X$ commutes with a normal operator $N$, show that it also commutes with $N^*$.  \\
\hint The entire function $f(z)=e^{zN^*}Xe^{-zN^*}=
e^{zN^*}e^{-\ol{z}N}Xe^{\ol{z}N}e^{-zN^*}$ is bounded.

\item Show that if $M$ and $N$ are normal and $MX=XN$, then
$M^*X=XN^*$.  \\
\hint $\begin{sbmatrix}M&0\\0&N\end{sbmatrix}$
commutes with $\begin{sbmatrix}0&X\\0&0\end{sbmatrix}$.

\item Show that two similar normal operators are unitarily equivalent. 
\hint take the unitary part of the polar decomposition of the 
similarity and apply part (a).
\end{parts}

\item Let $\A$ be a finite dimensional C*-algebra.
\begin{parts}
\item Let $\pi$ be an irreducible representation of $\A$ on $\B(H)$. Prove that $\dim H < \infty$, that $\pi(\A) = \B(H)$ and that $\ker\pi$ is a maximal ideal.
\item Show that $\A$ has a faithful finite dimensional representation.
\item Prove that $\A$ is $*$-isomorphic to the direct sum of a finite number of full matrix algebras.
\end{parts}


\end{Ex}
\end{document}
 

 