% LaTeX file -- PM 810, Winter 2019, Assignment 4

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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 810 \hfill Assignment 4 \hfill
Due Thursday March 28.}} 
\vskip 2em 

\begin{Ex}

\item 
\begin{parts}
\item For a von Neumann algebra $\N$, show that the extreme points of
$\ol{b_1(\N_+)}$ are precisely the projections. 
\item Show that for $\N=\B(H)$, the set of extreme
points of the unit ball is strictly larger than the set of unitaries.
\end{parts}

\item Suppose that $\rho$ is a representation of a C*-algebra $A$ on a Hilbert space $H$ with a unit cyclic vector $x$.
Define a state by $f(a)= \ip{\rho(a)x,x}$.  
Show that $\rho$ is unitarily equivalent to the GNS construct $\pi_f$ via a unitary
operator $U$ such that $Ux_f=x$.  
This shows that the GNS construction is unique.  
\hint  set $U\dot{a}=\rho(a)x$.


\item Suppose that $\pi$ and $\sigma$ are irreducible representations of a C*-algebra $\fA$.
Suppose that there is a non-zero operator $T$ such that $\pi(a)T=T\sigma(a)$ for all $a\in\fA$.
Prove that $\pi$ and $\sigma$ are unitarily equivalent. 
\hint show that the partial isometry in the polar decomposition of $T$ is the desired unitary.


\item An operator algebra $\A \subset \B(H)$ is \textit{reflexive} if every $T\in\B(H)$
satisfying $TM \subset M$ for every invariant subspace $M$ of $\A$ belongs to $\A$.
An operator $A$ is \textit{reflexive} if the WOT-closed (nonself-adjoint, unital) algebra $W(A)$
generated by $A$ is reflexive.
\begin{parts}
\item Prove that every von Neumann algebra is reflexive.

\item Prove that normal operators on separable Hilbert space are reflexive.\\
\hint $W(N) \subset W^*(N) \simeq L^\infty(\mu)$.
If $h(N) \in W^*(N) \setminus W(N)$, use Hahn--Banach to find a separating
functional $f$ in $L^1(\mu)$.  Factor $f = g\ol{k}$ where $g,k\in L^2(\mu)$.
\end{parts}


\item  Use the polar decomposition of a compact operator $K$ to show that it may be written as 
$K=\sum_{n\ge 1}s_n e_n f_n^*$ where $s_n$ is a positive sequence decreasing to 0, and $\{e_n\}$ and $\{f_n\}$ are
orthonormal sequences.  Here $ef^*$ is the rank one operator $ef^*(x)= \ip{x,f}e$. 
The sequence $s_n(K)$ are called the \textbf{singular values} of $K$.


\item A compact operator $K$ is trace class if $\|K\|_1:=\sum_{n\ge1}s_n(K) <\infty$.
The collection of all trace class operators on a separable Hilbert space $H$ is denoted by $\fS_1$.
\begin{parts}
\item Show that if $x_n$ and $y_n$ are orthonormal sequences, then $\sum_{n\ge1}|\ip{Kx_n,y_n}| \le \|K\|_1$.

\item Show that $\fS_1$ is complete subspace in the trace norm, and that the ideal of finite rank operators is dense in $\fS_1$.

\item Show that $\|AKB\|_1\le \|A\|\,\|K\|_1\,\|B\|$.  Hence  $\fS_1$ is a non-closed 2-sided ideal of $\B(H)$.

\item Fix an orthonormal basis $\{e_n\}$ and define the trace by $\Tr(K)=\sum_{n\ge 1} \ip{Ke_n,e_n}$. 
Show that $\Tr(KT)=\Tr(TK)$ for all $K\in\fS_1$ and $T\in\B(H)$.  
Hence deduce that $\Tr$ is independent of the choice of basis.

\item Each $T\in\B(H)$ defines a linear functional $\phi_T$ on $\fS_1$ by $\phi_T(K)=\Tr(TK)$.  
Show that $\|\phi_T\|=\|T\|$.

\item Show that if $\phi$ is a linear functional on $\fS_1$, then the sesquilinear form 
$\langle x,y\rangle:=\phi(xy^*)$ determines a bounded linear operator $T$ such that $\phi=\phi_T$.

\item Deduce that $\B(H)$ is the dual space of $\fS_1$.  Show that this weak-$*$ topology on $\B(H)$ corresponds to the ultraweak topology.
\end{parts}


\end{Ex}
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