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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\makeheading{Functional Analysis 7212 \hfill Homework problem list}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:ContinuityOfSpectralRadius}
Suppose $A$ is a unital Banach algebra and fix $a,b\in A$.
\begin{enumerate}
\item
Show that $1\notin \sp_A(ab)$ if and only if $1\notin\sp_A(ba)$ using the identity $(1-ba)^{-1} = 1+ b(1-ab)^{-1}a$.
Deduce that $\sp_A(ab)\cup\{0\} = \sp_A(ba)\cup \{0\}$.
\item
Show that for any Banach subalgebra $B\subseteq A$ with $1_A\in B$,
for every $a\in B$, the spectral radius in $B$ of $a$ is equal to the spectral radius in $A$ of $a$, i.e., $r_B(a)=r_A(a)$.
\item
Suppose $a,b\in A$ commute.
Prove that $r(ab)\leq r(a)r(b)$ and $r(a+b)\leq r(a)+r(b)$.
\\
\emph{Hint: By (2), this computation can be performed in the unital commutative Banach subalgebra $B\subseteq A$ generated by $a$ and $b$.
In $B$, there is a helpful characterization of the spectrum.}
\item
Deduce from part (3) that if $A$ is commutative, the spectral radius $r:A\to [0,\infty)$ is continuous.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:SpectrumConvergence}
Let $A$ be a unital Banach algebra.
Suppose we have a norm convergent sequence $(a_n)\subset A$ with $a_n \to a$.
Prove that for every open neighborhood $U$ of $\sp(a)$, there is an $N>0$ such that $\sp(a_n)\subset U$ for all $n>N$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $A\in M_n(\bbC)$.
\begin{enumerate}
\item
As best as you can, describe $f(A)$ where $f\in \mathcal{O}(\sp(A))$.
\\
\emph{Hint: First consider the case that $A$ is a single Jordan block.}
\item
Determine as best you can which matrices $A\in M_n(\bbC)$ have square roots, i.e., when there is a $B\in M_n(\bbC)$ such that $B^2 = A$.
\\
\emph{Note: Such a $B$ is not necessarily unique.}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:SpectralCharacterizationsOfNormalElements}
Suppose $A$ is a C*-algebra and $a\in A$ is normal.
\begin{enumerate}
\item
Show $a$ is self-adjoint if and only if $\sp(a)\subset \mathbb{R}$.
\item
Show $a$ is unitary if and only if $\sp(a)\subset \mathbb{T}$.
\item
Show $a$ is a projection if and only if $\sp(a)\subset \{0,1\}$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:PositiveCones}
Let $A$ be a C*-algebra.
\begin{enumerate}
\item
Show that the following are equivalent for a self-adjoint $a\in A$:
\begin{enumerate}
\item
$\sp(a)\subset [0,\infty)$,
\item
For all $\lambda\geq \|a\|$, $\|a-\lambda\|\leq \lambda$, and
\item
There is a $\lambda\geq \|a\|$ such that $\|a-\lambda\|\leq \lambda$.
\end{enumerate}
For now, we will call such elements \emph{spectrally positive}.
\\
\emph{Note: It is implicit here that a spectrally positive element is self-adjoint.}
\item
Deduce that the spectrally positive elements in a C*-algebra form a closed cone, i.e., $A_+ = \set{a\in A}{a\geq 0}$ is closed, and for all $\lambda\in [0,\infty)$ and $a,b\in A_+$, we have $\lambda a+b \in A_+$.
\item
Show $a$ is positive ($a=b^*b$ for some $b$) if and only if $a$ is spectrally positive ($a=a^*$ and $\sp(a)\subset [0,\infty)$).
\\
\emph{Hint:
First, if $\sp(a)\subset [0,\infty)$, we can define $a^{1/2}$ via the continuous functional calculus.
Now suppose $a=b^*b$ for some $b\in B$.
Use the continuous functions $r\mapsto \max\{0,z\}$ and $r\mapsto -\min\{0, z\}$ on $\sp(a)$ to write $a=a_+-a_-$ where $\sp(a_\pm)\subset [0,\infty)$ and $a_+a_-=a_-a_+=0$.
Now look at $c=ba_-$.
Prove that $\sp(c^*c)\subset (-\infty, 0]$ and $\sp(cc^*) \subset [0,\infty)$ using part (1) of this problem.
Use part (1) of Problem \ref{prob:ContinuityOfSpectralRadius} to deduce that $c^*c=0$.
Finally, deduce $a_-=0$, and thus $a=a_+$.
}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:PositiveOrder}
For $a,b\in A$, we say $a\leq b$ if $b-a\geq 0$.
\begin{enumerate}
\item
Show that $\leq$ is a partial order.
\item
Show that if $a\leq b$, then for all $c\in A$, $c^*ac \leq c^*bc$.
\item
Suppose $0\leq a\leq b$.
Prove that $\|a\|\leq \|b\|$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:PositiveSquareRoot}
Let $A$ be a C*-algebra.
By the hint to part (4) of Problem \ref{prob:SpectralCharacterizationsOfNormalElements} that for $a\geq 0$, we can define an $a^{1/2}\geq 0$ such that $(a^{1/2})^2 = a$.
\begin{enumerate}
\item
Show that if $b\geq 0$ such that $b^2 = a$, then $b=a^{1/2}$.
\item
Prove that if $0\leq a\leq b$, then
$a^{1/2}\leq b^{1/2}$.
\item
Prove that if $0< a$ ($0\leq a$ and $a$ is invertible), then $00$ ($T\geq 0$ and $T$ is invertible) if and only if $B_T$ is positive definite, and $H$ is complete in the norm $\|\xi\|_T := B_T(\xi,\xi)^{1/2}$.
\\
\emph{Hint:
When $B_T$ is positive definite and $H$ is complete for $\|\cdot\|_T$, apply part (d) and look at the isometry $(H, \|\cdot\|_T) \to (H, \|\cdot\|)$ by $\xi\mapsto T^{1/2}\xi$. }
\end{enumerate}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Challenge!]
Suppose $H$ is a Hilbert space.
A \emph{quadratic form} on $H$ is a function $q: H\to \bbC$ such that:
\begin{enumerate}
\item
(quadratic)
$q(\lambda \xi) = |\lambda|^2 q(\xi)$ for all $\lambda\in \bbC$ and $\xi\in H$,
\item
(parallelogram identity)
$q(\eta+\xi)+q(\eta - \xi) = 2(q(\eta)+q(\xi))$
for all $\eta,\xi\in H$, and
\item
(continuous)
There is a $C>0$ such that
$|q(\xi)|\leq C\|\xi\|^2$ for all $\xi\in H$.
\end{enumerate}
Prove that
$$
(\eta,\xi) := \frac{1}{4}\sum_{k=0}^3 i^k q(\eta+i^k\xi)
$$
is a bounded sesquilinear form on $H$ such that $q(\xi)=(\xi,\xi)$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
For a Hilbert space $H$, we can define the \emph{conjugate} Hilbert space $\overline{H}=\set{\overline{\xi}}{\xi\in H}$ which has the conjugate vector space structure $\lambda \overline{\xi}+\overline{\eta} = \overline{\overline{\lambda} \xi + \eta}$ and the conjugate inner product $\langle \overline{\eta}, \overline{\xi}\rangle_{\overline{H}} = \langle \xi, \eta\rangle_H$.
\begin{enumerate}
\item
Prove that $\overline{H}$ is a Hilbert space.
\item
For $T\in B(H,K)$, define $\overline{T} : \overline{H} \to \overline{K}$ by $\overline{T}\overline{\xi} = \overline{T\xi}$.
Prove that $\overline{T}\in B(\overline{H}, \overline{K})$, and $\|T\| = \|\overline{T}\|$.
\item
Prove that $\overline{\,\cdot\,}$ is an endofunctor on the the category ${\sf Hilb}$ of Hilbert spaces with bounded operators
($\overline{\,\cdot\,}$ is a functor ${\sf Hilb}\to {\sf Hilb}$).
\item
For each $H\in {\sf Hilb}$, construct a linear isometry $u_H$ of $H^*$ onto $\overline{H}$ satisfying $u_H T^t = \overline{T} u_H$ for all $T\in B(H,K)$ where $T^t\in B(K^*, H^*)$ is the Banach adjoint of $T$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
For $T\in B(H)$, we define its \emph{numerical radius} as
$$
R(T) :=\sup_{\|\xi\|\leq 1}|\langle T\xi, \xi\rangle|.
$$
Prove that $r(T)\leq R(T)\leq \|T\| \leq 2R(T)$.
Deduce that if $T$ is normal, then $\|T\|=R(T)$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $A$ be a C*-algebra.
An element $u\in A$ is called a \emph{partial isometry} if $u^*u$ is a projection.
\begin{enumerate}
\item
Show that the following are equivalent:
\begin{enumerate}
\item
$u$ is a partial isometry.
\item
$u=uu^*u$.
\item
$u^*=u^*uu^*$.
\item
$u^*$ is a partial isometry.
\end{enumerate}
\emph{Hint: For $(a)\Rightarrow (b)$, apply the C*-axiom to $u-uu^*u$.}
\item
We say two projections $p,q\in A$ are \emph{(Murray-von Neumann) equivalent}, denoted $p\approx q$, if there is a partial isometry $u\in A$ such that $uu^*=p$ and $u^*u=q$.
Prove that $\approx$ is an equivalence relation on $P(A)$, the set of projections of $A$.
\item
Describe the set of equivalence classes $P(A)/\approx$ for $A=B(\ell^2)$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $x=u|x|$ is the polar decomposition of $x\in B(H)$.
Show that $x^*=u^*|x^*|$ is the polar decomposition.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{prob}
%Suppose $A$ is a C*-algebra, and denote the unitization of $A$ by $\widetilde{A}$.
%\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[MO:325725]
Suppose $A$ is a unital C*-algebra and $I\leq A$ is an ideal.
Let $q: A\to A/I$ be the canonical surjection.
\begin{enumerate}
\item
Show that unital $*$-homomorphisms $C[0,1] \to A$ are in canonical bijection with positive elements of $A$ with norm at most 1.
\item
Show that if $a+I \in A/I$ is positive with norm at most 1, there is a positive $\widetilde{a} \in A$ with norm at most 1 such that $\widetilde{a} + I = a + I$.
\\\emph{Hint: Since $\sp_{A/I}(a+I) \subseteq \sp_A(a)$, $f(q(a))=q(f(a))$ and thus $f(a+I)=f(a)+I$ for all $f\in C(\sp_A(a))$.
Now pick $f$ carefully.
%
%Here is one way to approach this.
%\begin{enumerate}
%\item
%Denote the unitization of $A/I$ by $\widetilde{A/I}$.
%Show that $\sp_{\widetilde{A/I}}(a+I)\subseteq \sp_{A}(a)$.
%\item
%Show for all polynomials $p$ such that $p(0)=0$, $p(a+I)=p(a)+I$.
%\item
%Show that for any $f\in C(\sp_{\widetilde{A/I}}(a+I))$ such that $f(0)=0$, $f(a+I)\in A/I$.
%\item
%Show that for any $f\in C(\sp_{\widetilde{A/I}}(a+I))$ such that $f(0)=0$, $f(a+I)=f(a)+I$.
%\item
%Define
%$$
%f(t):= \begin{cases}
%t &\text{if }0\leq t\leq 1
%\\
%1 &\text{if }t>1.
%\end{cases}
%$$
%Show that $\widetilde{a}:= f(a)$ is such a desired lift of $a$.
%\end{enumerate}
}
\item
Deduce that for every unital $*$-homomorphism $\phi: C[0,1] \to A/I$, there is a unital $*$-homomorphism $\widetilde{\varphi}: C[0,1] \to A$ with $\phi = q \circ \widetilde{\phi}$.
\item
Discuss the connection between the above statement and the Tietze Extension Theorem when $A$ is commutative.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $H$ be a Hilbert space.
Compute the extreme points of the unit balls of
\begin{enumerate}
\item $\cK(H)$,
\item $\cL^1(H)$, and
\item $B(H)$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{prob}[Noncommutative torus]
%In this problem, we will construct the non-commutative torus/irrational rotation algebra in two ways.
%Consider $e^{2\pi i \alpha}\in \bbT$ where $\theta [0,1)$.
%\begin{enumerate}
%\item
%Define $U, V\in B(L^2(\bbT))$ by $(Uf)(e^{2\pi i \alpha})=e^{2\pi i \alpha}f(e^{2\pi i \alpha})$ and $(Vf)(e^{2\pi i \alpha})= f(e^{2\pi i (\alpha-\theta)})$.
%\begin{enumerate}
%\item
%Show that $U,V$ are unitary and $UV = e^{2\pi i \alpha}VU$.
%\item
%Let $A$ be the unital C*-subalgebra of $B(L^2(\bbT))$ generated by $U$ and $V$.
%Show that when $\theta = 0$, then $A$ is commutative, and $\widehat{A} \cong \bbT^2$.
%\end{enumerate}
%\item
%\item
%%Show that the transformation $T: \bbT \to \bbT$ given by $\exp(2\pi i\alpha)\mapsto \exp(2\pi i (\alpha+\theta))$ is ergodic, i.e., if $E\subseteq \bbT$ such that $\mu(E) = \mu(T^n(E))$ for all $n\in \bbZ$, then $m(E)\in \{0,1\}$ (here $m$ is Lebesgue measure on $\bbT$).
%%Deduce that if $f\in L^\infty(\bbT)$ such that $f=f\circ T^{-1}$, then $f=\lambda 1_\bbT$.
%\end{enumerate}
%
%\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $H$ be a Hilbert space.
Prove that the trace $\operatorname{Tr}$ induces isometric isomorphims:
\begin{enumerate}
\item $\cK(H)^* \cong \cL^1(H)$, and
\item $\cL^1(H)^*\cong B(H)$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $H$ is a Hilbert space and $K\subseteq H$ is a closed subspace.
Let $p_K\in B(H)$ be associated orthogonal projection onto $K$.
\begin{enumerate}
\item
Suppose $x\in B(H)$.
Prove that:
\begin{enumerate}
\item $xK\subseteq K$ if and only if $xp_K= p_K xp_K$.
\item $x^*K\subseteq K$ if and only if $p_Kx = p_K x p_K$.
\item $xK\subseteq K$ and $x^*K\subseteq K$ if and only if $[x,p_K]=0$.
\end{enumerate}
\item
Prove that if $M\subseteq B(H)$ is a $*$-closed subalgebra, then $MK\subseteq K$ if and only if $p_K\in M'$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:Amplification}
Suppose $H$ is a Hilbert space.
\begin{enumerate}
\item
Suppose $K$ is another Hilbert space.
Define the tensor product Hilbert space $H\overline{\otimes} K$ by completing the algebraic tensor product vector space $H\otimes K$ in the 2-norm associated to the sesquilinear form $\langle \eta \otimes \xi , \eta'\otimes \xi'\rangle := \langle \eta, \eta'\rangle \langle \xi, \xi'\rangle$.
Find a unitary isomorphism $H\overline{\otimes} K \cong \bigoplus_{i=1}^{\dim K} H$.
\item
Find a unital $*$-isomorphism $B(\bigoplus_{i=1}^n H) \cong M_n(B(H))$.
\\
\emph{Hint: use orthogonal projections.}
\item
Suppose $S\subseteq B(H)$, and let $\alpha : B(H) \to M_n(B(H))$ be the amplification
$$
x\longmapsto
\begin{pmatrix}
x
\\
&\ddots
\\ && x
\end{pmatrix}.
$$
Prove that:
\begin{enumerate}
\item
$\alpha(S)' = M_n(S')$, and
\item
If $0,1\in S$, then $M_n(S)' = \alpha(S')$.
\item
Deduce that when $0,1\in S$, $\alpha(S)''=\alpha(S'')$.
\end{enumerate}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $(X,\mu)$ be a $\sigma$-finite measure space, and consider the map $M: L^\infty(X,\mu) \to B(L^2(X,\mu))$ given by $(M_f\xi)(x) = f(x)\xi(x)$ for $\xi\in L^2(X,\mu)$.
\begin{enumerate}
\item
Prove that $M$ is an isometric unital $*$-homomorphism.
\item
Let $A\subset B(L^2(X,\mu))$ be the image of the map $M$.
Prove that $A = A'$.
\\
\emph{Hint: If you're stuck with (2), try the case $X = \bbN$ with counting measure.}
%If you're still stuck, try the case $(X,\mu)$ is a finite measure space, so $1\in L^2(X,\mu)$.}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $H$ be a Hilbert space.
The \emph{weak operator topology (WOT)} on $B(H)$ is the topology induced by the separating family of seminorms $T\mapsto |\langle T\eta, \xi\rangle|$ for $\eta, \xi \in H$.
The \emph{strong operator topology (SOT)} on $B(H)$ is induced by the separating family of seminorms $x\mapsto \|T\xi\|_H$ for $\xi \in H$.
\begin{enumerate}
\item
Prove that every WOT open set is SOT open.
Equivalently, prove that if a net $(T_\lambda)_{\lambda\in\Lambda}\subset B(H)$ converges to $T\in B(H)$ SOT, then $T_\lambda \to T$ WOT.
\item
Prove that the WOT is equal to the SOT on $B(H)$ if and only if $H$ is finite dimensional.
\item
Show that the following are equivalent for a linear functional $\varphi$ on $B(H)$:
\begin{enumerate}[(a)]
\item
There are $\eta_1,\dots, \eta_n, \xi_1,\dots, \xi_n \in H$ such that $\varphi(T) = \sum_{i=1}^n \langle T \eta_i, \xi_i \rangle$.
\item
$\varphi$ is WOT-continuous.
\item
$\varphi$ is SOT-continuous.
\end{enumerate}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:CyclicSeparating}
Suppose $M\subset B(H)$ is a unital $*$-subalgebra.
A vector $\xi \in H$ is called:
\begin{itemize}
\item
\emph{cyclic} for $M$ if $M\xi$ is dense in $H$.
\item
\emph{separating} for $M$ if for every $x,y\in M$, $x\xi = y\xi$ implies $x=y$.
\end{itemize}
\begin{enumerate}
\item
Prove that $\xi$ is cyclic for $M$ if and only if $\xi$ is separating for $M'$.
\item
Prove that $H$ can be orthogonally decomposed into $M$-invariant subspaces $H=\bigoplus_{i\in I} K_i$, such that each $K_i$ is cyclic for $M$ (has a cyclic vector).
Prove that if $H$ is separable, this decomposition is countable.
\item
Prove that if $M$ is abelian and $H$ is separable, then there is a separating vector in $H$ for $M$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:IncreasingNet}
Suppose $H$ is a Hilbert space, and $(x_\lambda)$ is an increasing net of positive operators in $B(H)$ which is bounded above by the positive operator $x\in B(H)$, i.e., $\lambda\leq \mu$ implies $x_\lambda \leq x_\mu$, and $0\leq x_\lambda \leq x$ for all $\lambda$.
Prove that the following are equivalent.
\begin{enumerate}
\item
$x_\lambda \to x$ SOT.
\item
$x_\lambda \to x$ WOT.
\item
For every $\xi \in H$, $\omega_\xi(x_\lambda) = \langle x_\lambda \xi,\xi\rangle \nearrow \langle x\xi,\xi\rangle = \omega_\xi(x)$.
\item
There exists a dense subspace $D\subset H$ such that for every $\xi \in D$, $\omega_\xi(x_\lambda) = \langle x_\lambda \xi,\xi\rangle \nearrow \langle x\xi,\xi\rangle = \omega_\xi(x)$.
\end{enumerate}
We say an increasing net of positive operators $(x_\lambda)$ \emph{increases to} $x\in B(H)_+$, denoted $x_\lambda \nearrow x$, if any of the above equivalent conditions hold.
\\
\emph{Hint: Show it suffices to prove $(3)\Rightarrow (1)$ and $(4)\Rightarrow (3)$.
Try proving these implications.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $H$ be a Hilbert space and let $T\in B(H)$.
Prove that the following are equivalent.
(You may use any results from last semester that you'd like without proof.)
\begin{enumerate}
\item
$T$ is compact and normal.
\item
$T$ has an orthonormal basis of eigenvectors $(e_i)_{i\in I}$ such that the corresponding eigenvalues $\lambda_i\to 0$, with at most countably many of the $\lambda_i \neq 0$.
\item
There is a countable orthonormal subset $(\xi_n)_{n\in \bbN}\subset H$ and a sequence $(\lambda_n)\subset \bbC$ such that $\lambda_n \to 0$ and $T = \sum_{n\in \bbN} \lambda_n |\xi_n\rangle\langle \xi_n |$, which converges in operator norm.
\item
There is a sequence $(\lambda_n)\subset \bbC$ such that $\lambda_n \to 0$ and a countable family of finite rank projections $E_n \subset B(H)$ such that $T = \sum_{n\in \bbN} \lambda_n E_n$, which converges in operator norm.
\item
There is a discrete set $X$ equipped with counting measure $\nu$, a function $f\in c_0(X)$, and a unitary $U\in B(\ell^2 X, H)$ such that $T = UM_fU^*$ where $M_f \xi = f\xi$ for $\xi \in \ell^2X$.
\\
\emph{Note: $U\in B(K,H)$ is unitary if $UU^* = \operatorname{id}_H$ and $U^*U=\operatorname{id}_K$.}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:CompletelyPositive}
Suppose $A$ is a unital C*-algebra.
A linear map $\Phi: A \to B(H)$ is called \emph{completely positive} if for every $a=(a_{i,j})\geq 0$ in $M_n(A)$, $(\Phi(a_{i,j})) \geq 0$ in $M_n(B(H))\cong B(H^n)$.
Such a map is \emph{unital} if $\Phi(1) = 1$.
\begin{enumerate}
\item
Show that $\langle x \otimes \eta, y \otimes \xi\rangle := \langle \Phi(y^*x)\eta, \xi\rangle_H$ on $A\otimes H$ linearly extends to a well-defined positive sesquilinear form.
\item
Show that for $V$ a vector space with positive sesquilinear form $B(\,\cdot\,,\,\cdot\,)$, $N_B = \set{v\in V}{B(v,v) = 0}$ is a subspace of $V$, and $B$ descends to an inner product on $V/N_B$.
\item
Define $K$ to be completion of $(A\otimes H)/N_{\langle\,\cdot\,,\,\cdot\,\rangle}$ in $\|\cdot\|_2$.
Find a unital $*$-homormophism $\Psi: A \to B(K)$, and an isometry $v \in B(H,K)$ such that $\Phi(m) = v^* \Psi(m)v$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $y\in B(H)$ is positive.
\begin{enumerate}
\item
Show that if $y\notin K(H)$,
then there is a $\lambda>0$ and a projection $p$ with infinite dimensional range such that $y\geq \lambda p$.
%then there is a diagonalizable $0\leq z\leq y$ such that $z\notin K(H)$.
%Here, we say $z$ is \emph{diagonalizable} if $H$ is the orthogonal direct sum of eigenspaces of $z$.
%(Alternatively, we can write $z=\sum \lambda_n p_n$ where $(\lambda_n)$ is a sequence of numbers and $(p_n)$ is a sequence of mutually orthogonal projections such that $\sum p_n = 1$.
%Here, both sums converge pointwise, a.k.a.~SOT.)
\item
Deduce that if $x\mapsto\operatorname{Tr}(xy)$ is bounded on $\cL^p(H)$ where $1\leq p<\infty$, then $y\in K(H)$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $A\subseteq B(H)$ is a unital C*-subalgebra and $\xi \in H$ is a cyclic vector for $A$.
Consider the vector state $\omega_\xi = \langle \,\cdot\,\xi,\xi\rangle$.
Prove there is a bijective correspondence between:
\begin{enumerate}
\item
positive linear functionals $\varphi$ on $A$ such that $0\leq \varphi \leq \omega_\xi$ ($\omega_\xi - \varphi \geq 0$), and
\item
operators $0\leq x \leq 1$ in $A'$.
\end{enumerate}
\emph{Hint:
For $0\leq x\leq 1$ in $A'$, define $\varphi_x(a):= \langle ax\xi,\xi\rangle$ for $a\in A$.
(Why is $0\leq \varphi_x\leq \omega_\xi$?)
For the reverse direction, use the bijective correspondence between sesquilinear forms and operators.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\mbox{}
\begin{enumerate}
\item
Prove that a unital $*$-subalgebra $M\subseteq B(H)$ is a von Neumann algebra if and only if its unit ball is $\sigma$-WOT compact.
\item
Let $M\subset B(H)$ be a von Neumann algebra and $\Phi: M \to B(K)$ a unital $*$-homomorphism.
Deduce that if $\Phi$ is $\sigma$-WOT continuous and injective, then $\Phi(M)$ is a von Neumann subalgebra of $B(K)$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $X$ is a compact Hausdorff topological space and $E: (X, \cM) \to B(H)$ is a Borel spectral measure.
Prove that the following conditions are equivalent.
\begin{enumerate}
\item
$E$ is regular, i.e., for all $\xi \in H$, $\mu_{\xi,\xi}(S) = \langle E(S) \xi, \xi\rangle$ is a finite regular Borel measure.
\item
For all $S\in \cM$, $E(S) = \sup \set{E(K)}{K \text{ is compact and }K\subseteq S}$.
\item
For all $S\in \cM$, $E(S) = \inf \set{E(U)}{U \text{ is open and } S\subseteq U}$
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $x\in B(H)$ is normal.
Show that $\chi_{\{0\}}(x)=p_{\ker(x)}$ and $\chi_{\sp(x)\setminus\{0\}} = p_{\overline{xH}}$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $H$ be a separable Hilbert space and $A\subseteq B(H)$ an abelian von Neumann algebra.
Prove that the following are equivalent.
\begin{enumerate}
\item
$A$ is maximal abelian, i.e., $A=A'$.
\item
$A$ has a cyclic vector $\xi\in H$.
\item
For every norm separable SOT-dense C*-subalgebra $A_0\subset A$, $A_0$ has a cyclic vector.
\item
There is a norm separable SOT-dense C*-subalgebra $A_0\subset A$ such that $A_0$ has a cyclic vector.
\item
There is a finite regular Borel measure $\mu$ on a compact Hausdorff second countable space $X$ and a unitary $u\in B(L^2(X,\mu),H)$ such that $f\mapsto uM_fu^*$ is an isometric $*$-isomorphism $L^\infty(X,\mu) \to A$.
\end{enumerate}
\emph{Hints:
\\
For $(1)\Rightarrow (2)$, use Problem \ref{prob:CyclicSeparating}.
\\
For $(3)\Rightarrow (4)$ it suffices to construct a norm separable SOT-dense C*-algebra.
First show that $A_* = \cL^1(H)/A_\perp$ is a separable Banach space.
Then show that $A$ is $\sigma$-WOT separable, which implies SOT-separable.
Take $A_0$ to be the unital C*-algebra generated by an SOT-dense sequence.
\\
For $(4)\Rightarrow (5)$ show that $A_0$ separable implies $X=\widehat{A}_0$ is second countable.
Define $\mu=\mu_{\xi,\xi}$ on $X$, and show that the map $C(X)\to H$ by $f \mapsto \Gamma^{-1}(f)\xi$ is a $\|\cdot\|_2-\|\cdot\|_H$ isometry with dense range.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $E: (X, \cM) \to P(H)$ is a spectral measure with $H$ separable, and let $A\subset B(H)$ be the unital C*-algebra which is the image of $L^\infty(E)$ under $\int \cdot\, dE$.
Suppose there is a cyclic unit vector $\xi \in H$ for $A$.
\begin{enumerate}
\item
Show that $\omega_\xi(f)= \langle (\int f dE) \xi , \xi\rangle$ is a faithful state on $L^\infty(E)$ ($\omega_\xi(|f|^2)=0 \Longrightarrow f=0$).
\item
Consider the finite non-negative measure $\mu = \mu_{\xi,\xi}$ on $(X,\cM)$.
Show that a measurable function $f$ on $(X,\cM)$ is essentially bounded with respect to $E$ if and only if $f$ is essentially bounded with respect to $\mu$.
\item
Deduce that for essentially bounded measurable $f$ on $(X,\cM)$, $\|f\|_E = \|f\|_{L^\infty(X, \cM,\mu)}$.
\item
Construct a unitary $u\in B(L^2(X,\cM,\mu), H)$ such that for all $f\in L^\infty(E)=L^\infty(X, \cM, \mu)$, $(\int f dE) u =u M_f$.
\item
Deduce that $A\subset B(H)$ is a maximal abelian von Neumann algebra.
\end{enumerate}
%\emph{Remark: We saw in class that the map $\int \cdot \, dE : L^\infty(E) \to A$ is
%\begin{itemize}
%\item
%\emph{normal}: for $0\leq f_\lambda \nearrow f$ in $L^\infty(E)$, $\int f_\lambda dE \nearrow \int f dE$.
%\end{itemize}
%We will see in a later homework problem that this condition is equivalent to $\sigma$-WOT continuity.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $H$ is a separable infinite dimensional Hilbert space.
Prove that $K(H)\subset B(H)$ is the unique norm closed 2-sided proper ideal.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Classify all abelian von Neumann algebras $A\subset B(H)$ when $H$ is separable.
\\
\emph{Hint: Use a maximality argument to show you can write $1 = p + q$ with $p,q\in P(A)$ such that $q$ is diffuse and $p=\sum p_i$ (SOT) with all $p_i$ minimal.
Then analyze $Aq$ and $Ap$.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $M\subseteq B(H)$ is a von Neumann algebra and $p,q\in P(M)$.
Define $p\wedge q \in B(H)$ to be the orthogonal projection onto $pH\cap qH$.
Prove that $p \wedge q \in M$ two separate ways:
\begin{enumerate}
\item
Show that $pH \cap qH$ is $M'$-invariant, and deduce $p\wedge q \in M$.
\item
Show that $p\wedge q$ is the SOT-limit of $(pq)^n$ as $n\to \infty$.
\\\emph{Hint:
You could proceed as follows, but a quicker proof would be much appreciated!
\begin{enumerate}
\item
Use (2) of Problem \ref{prob:PositiveOrder} to show $(pq)^n p$ is a decreasing sequence of positive operators.
\item
Show $(pq)^n p$ converges SOT to a positive operator $x\in M$.
%(A candidate for the SOT limit can be found by looking at the quadratic form $q(\xi)=\langle \xi, \xi \rangle = \inf_n \langle (pq)^np \xi, \xi\rangle$ and defining a bounded sesquilinear form via polarization.)
\item
Show that $x^2=x$, and deduce $x\leq p$ is an orthogonal projection.
\item
Show that $xqp = x$, and deduce $xqx = x$.
\item
Show that $x\leq q$, and deduce $x\leq p\wedge q$.
\item
Show that $(p\wedge q)(pq)^n$ converges SOT to both $p\wedge q$ and $x$, and deduce $x = p\wedge q$.
\item
Finally, show $(pq)^n$ converges SOT to $xq = p\wedge q$.
\end{enumerate}
}
\end{enumerate}
Define $p\vee q$ as the projection onto $\overline{pH+qH}$.
Show that $p\vee q \in M$ in two separate ways:
\begin{enumerate}
\item
Prove that $\overline{pH + qH}$ is $M'$-invariant, and deduce $p\vee q\in M$.
\item
Show that $p\vee q = 1-(1-p)\wedge (1-q)$ and use that $p\wedge q \in M$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $N\subseteq M\subset B(H)$ is a unital inclusion of von Neumann algebra and $p\in P(N)$.
\begin{enumerate}
\item
Prove that $(N'p)\cap pMp = (N'\cap M)p$.
\item
Deduce that if $p\in P(M)$, $Z(pMp) = Z(M)p$.
\item
Deduce that if $p\in P(M)$ and $M$ is a factor, then $pMp$ is a factor.
\item
Prove that when $M$ is a factor and $p\in P(M)$, the map $M' \to M'p$ by $x\mapsto xp$ is a unital $*$-algebra isomorphism.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:TypeI}
Prove that the following conditions are equivalent for a von Neumann algebra $M\subseteq B(H)$:
\begin{enumerate}
\item
Every non-zero $q\in P(M)$ majorizes an abelian projection $p\in P(M)$.
\item
$M$ is type {\rm I} (every non-zero $z\in P(Z(M))$ majorizes an abelian $p\in P(M)$).
\item
There is an abelian projection $p\in P(M)$ whose central support $z(p)=\bigvee_{u\in U(M)} u^*pu \in Z(M)$ is $1_M$.
\end{enumerate}
\emph{Hints:
\\
For $(2)\Rightarrow (3)$, if $p\in P(M)$ is abelian with $z(p)\neq 1$, then there is an abelian projection $q\in P(M)$ such that $z(q) \leq 1-z(p)$.
Show that $pMq = 0$ and $p+q$ is an abelian projection.
Now use Zorn's Lemma.
\\
For $(3)\Rightarrow (1)$, suppose $p\in P(M)$ is abelian with $z(p) = 1$ and $q\in P(M)$ is non-zero.
Show there is a non-zero partial isometry $u\in M$ such that $uu^* \leq p$ and $u^*u \leq q$.
Deduce that $uu^*$ is abelian, and then prove $u^*u$ is abelian.
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:TypeDecomposition}
Show that for every von Neumann algebra $M$, there are unique central projections $z_{\rm I}$, $z_{{\rm II}_1}$, $z_{{\rm II}_\infty}$, and $z_{\rm III}$ (some of which may be zero) such that
\begin{itemize}
\item
$Mz_{\rm I}$ is type ${\rm I}$, $Mz_{{\rm II}_1}$ is type ${\rm II}_1$, $Mz_{{\rm II}_\infty}$ is type ${\rm II}_\infty$, and $Mz_{\rm III}$ is type ${\rm III}$, and
\item
$z_{\rm I}+z_{{\rm II}_1}+z_{{\rm II}_\infty}+z_{\rm III}= 1$
\end{itemize}
\emph{Hint:
You could proceed as follows:
\begin{enumerate}
\item
First, show that if $M$ has an abelian projection $p$, then $z(p)$ is type ${\rm I}$.
Then use a maximality argument to construct $z_{\rm I}$.
For this, you could adapt the hint for $(2)\Rightarrow (3)$ in Problem \ref{prob:TypeI}.
\item
Replacing $M,H$ with $M(1-z_{\rm I}), (1-z_{\rm I})H$, we may assume $M$ has no abelian projections.
Show that if $M$ has a finite central projection $z$, then $Mz$ is type ${\rm II}_1$.
Now use a maximality argument to construct $z_{{\rm II}_1}$.
This hinges on proving the sum of two orthogonal finite central projections is finite.
(Proving this is much easier than proving the sup of two finite projections is finite!)
\item
By compression, we may now assume that $M$ has no abelian projections and no finite central projections.
Show that if $M$ has a nonzero finite projection $p$, then its central support $z(p)$ satisfies $Mz(p)$ is type ${\rm II}_\infty$.
Use a maximality argument to construct $z_{{\rm II}_\infty}$.
\item
Compressing one more time, we may assume $M$ has no finite projections, and thus $M$ is purely infinite and type ${\rm III}$.
\end{enumerate}
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:FiniteDimensionalVNA}
Let $M\subseteq B(H)$ be a finite dimensional von Neumann algebra.
\begin{enumerate}
\item
Prove $M$ has a minimal projection.
\item
Deduce that $Z(M)$ has a minimal projection.
\item
Prove that for any minimal projection $p\in Z(M)$, $Mp$ is a type ${\rm I}$ factor.
\item
Prove that $M$ is a direct sum of matrix algebras.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $H$ is infinite dimensional.
Prove that $B(H)$ does not admit a $\sigma$-WOT continuous tracial state.
\\
\emph{Optional:
Instead, prove that $B(H)$ does not admit a non-zero tracial linear functional.
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $M\subseteq B(H)$ and $N\subseteq B(K)$ are von Neumann algebras, and let $H\overline{\otimes} K$ be the tensor product of Hilbert spaces as in Problem \ref{prob:Amplification}.
\begin{enumerate}
\item
Show that for every $m\in M$ and $n\in N$, the formula $(m\otimes n)(\eta \otimes \xi) := m\eta \otimes n\xi$ gives a unique well-defined operator $m\otimes n \in B(H\overline{\otimes} K)$.
\item
Let $M\overline{\otimes} N = \set{m\otimes n}{m\in M, n\in N}''\subset B(H\overline{\otimes} K)$.
Show that the linear extension of the map from the algebraic tensor product $M\otimes N$ to $M\overline{\otimes}N$ given by $m\otimes n \mapsto m\otimes n$ is a well-defined injective unital $*$-algebra map onto an SOT-dense unital $*$-subalgebra.
\\
\emph{Hint for injectivity:
Suppose $x = \sum_{i=1}^k m_i\otimes n_i$ is not zero in $M\otimes N$.
Reduce to the case $\{n_1,\dots, n_k\}$ is linearly independent and all $m_i \neq 0$.
Show that for each $i=1,\dots, k$, there exists a $k_i>0$ and $\{\eta^i_j , \xi^i_j\}_{j=1}^{k_i}$ such that $\sum_{j=1}^{k_i}\langle n_{i'} \eta_j^i, \xi_j^i\rangle = \delta_{i=i'}$.
(Sub-hint:
Consider $F = \operatorname{span}_\bbC\{n_1,\dots, n_k\}\subset N$, a closed normed space, and look at $\Phi:H\times \overline{H} \to F^*$ by $(\eta, \xi) \mapsto \langle \,\cdot\,\eta, \xi\rangle$.
Show that $\operatorname{span}_\bbC(\Phi(H))= F^*$.)
Now pick $\kappa,\zeta\in H$ such that $\langle m_1\kappa, \zeta\rangle \neq 0$, and deduce $\sum_{j=1}^{k_1}\langle x (\kappa \otimes \eta^1_j), \zeta \otimes \xi^1_j\rangle_{H\overline{\otimes}K} \neq 0$.
}
\item
We denote by $B(H)\otimes 1$ the image of $B(H)$ under the map $x\mapsto x\otimes 1 \in B(H\overline{\otimes} K)$.
Prove that $B(H)\otimes 1$ is a von Neumann algebra.
\\
\emph{Hint: Show that $(B(H)\otimes 1)' = 1\otimes B(K)$.
Then by symmetry, $(1\otimes B(K))' = B(H)\otimes 1$ is a von Neumann algebra.}
\item
Prove that $B(H\overline{\otimes} K) = B(H)\overline{\otimes} B(K)$.
\\
\emph{Hint: Calculate the commutant of the image of the algebraic tensor product $(B(H)\otimes B(K))'=\bbC1$ and use (2).}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $S_\infty$ be the group of finite permutations of $\bbN$.
\begin{enumerate}
\item
Show that $S_\infty$ is ICC.
Deduce that $LS_\infty$ is a ${\rm II}_1$ factor.
\item
Give an explicit description of a projection with trace $k^{-n}$ for arbitrary $n,k\in\bbN$.
\\
\emph{Hint: Find such a projection in $\bbC S_\infty \subset L S_\infty$.}
\item
Find an increasing sequence $F_n \subset LS_\infty$ of finite dimensional von Neumann subalgebras such that $LS_\infty = \left(\bigcup_{n=1}^\infty F_n \right)''$.
\end{enumerate}
\emph{Note: A ${\rm II}_1$ factor which is generated by an increasing sequence of finite dimensional von Neumann subalgebras as in (3) above is called \emph{hyperfinite}.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $M$ be a von Neuman algebra.
Suppose $a,b\in M$ with $0\leq a\leq b$.
Prove there is a $c\in M$ such that $a = c^*bc$.
Deduce that a 2-sided ideal in a von Neumann algebra is \emph{hereditary}: $0\leq a \leq b \in M$ implies $a\in M$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $M$ be a factor.
Prove that if $M$ is finite or purely infinite, then $M$ is algebraically simple, i.e., $M$ has no 2-sided ideals.
\\
\emph{Note: You may use that a ${\rm II}_1$ factor has a (faithful $\sigma$-WOT continuous) tracial state.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:CompletelyAdditive}
A positive linear functional $\varphi \in M^*$ is called \emph{completely additive} if for any family of pairwise orthogonal projections $(p_i)$, $\varphi(\sum p_i) = \sum \varphi(p_i)$. (Here, $\sum p_i$ converges SOT.)
Suppose $\varphi, \psi \in M^*$ are completely additive and $p\in P(M)$ such that $\varphi(p)<\psi(p)$.
Then there is a non-zero projection $q\leq p$ such that $\varphi(qxq) < \psi(qxq)$ for all $x\in M_+$ such that $qxq\neq 0$.
\\
\emph{Hint:
Choose a maximal family of mutually orthogonal projections $e_i \leq p$ for which $\psi(e_i) \leq \varphi(e_i)$.
Consider $e = \bigvee e_i$, and show that $\psi(e)\leq \varphi(e)$.
Set $q = p-e$, and show that for all projections $r\leq q$, $\varphi(r) < \psi(r)$.
Then show $\varphi(qxq) < \psi(qxq)$ for all $x\in M_+$ such that $qxq\neq 0$.
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Show that the following conditions are equivalent for a positive linear functional $\varphi\in M^*$ for a von Neumann algebra $M$:
\begin{enumerate}
\item
$\varphi$ is $\sigma$-WOT continuous,
\item
$\varphi$ is \emph{normal}: $x_\lambda \nearrow x$ implies $\varphi(x_\lambda)\nearrow \varphi(x)$, and
\item
$\varphi$ is \emph{completely additive}: for any family of pairwise orthogonal projections $(p_i)$, $\varphi(\sum p_i) = \sum \varphi(p_i)$.
(Here, $\sum p_i$ converges SOT.)
%\\
%\emph{Hint: $\sum p_\lambda$ converges SOT if and only if $\sum p_\lambda$ converges WOT by Problem \ref{prob:IncreasingNet}.}
\end{enumerate}
\emph{Hint:
For $(3)\Rightarrow (1)$, show if $p\in P(M)$ is non-zero, then pick $\xi\in H$ such that $\varphi(p) < \langle p\xi,\xi\rangle$.
Use Problem \ref{prob:CompletelyAdditive} to find a non-zero $q\leq p$ such that $\varphi(qxq) < \langle xq\xi, q\xi\rangle$ for all $x\in M$.
Use the Cauchy-Schwarz inequality to show $x\mapsto \varphi(xq)$ is SOT-continuous, and thus $\sigma$-WOT continuous.
Now use Zorn's Lemma to consider a maximal family of mutually orthogonal projections $(q_i)_{i\in I}$ for which $x\mapsto \varphi(xq_i)$ is $\sigma$-WOT continuous.
Show $\sum q_i = 1$.
For finite $F\subseteq I$, define $\varphi_F(x) = \sum_{i\in F} \varphi(x q_i)$.
Ordering finite subsets by inclusion, we get a net $(\varphi_F)\subset M_*$.
Show that $\varphi_F \to \varphi$ in norm in $M^*$.
Deduce that $\varphi \in M_*$ since $M_* \subset M^*$ is norm-closed.
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:NormalHom}
Let $\Phi: M \to N$ be a unital $*$-homomorphism between von Neumann algebras.
\begin{enumerate}
\item
Prove that the following two conditions are equivalent:
\begin{enumerate}
\item
$\Phi$ is \emph{normal}: $x_\lambda \nearrow x$ implies $\Phi(x_\lambda) \nearrow \Phi(x)$.
\item
$\Phi$ is $\sigma$-WOT continuous.
\end{enumerate}
\item
Prove that if $\Phi$ is normal, then $\Phi(M)\subset N$ is a von Neumann subalgebra.
\\
\emph{Hint: $\ker(\Phi)\subset M$ is a $\sigma$-WOT closed 2-sided ideal.}
\item
Let $\varphi$ be a normal state on a a von Neumann algebra $M$, and let $(H_\varphi, \Omega_\varphi, \pi_\varphi)$ be the cyclic GNS representation of $M$ associated to $\varphi$, i.e., $H_\varphi = L^2(M,\varphi)$, $\Omega_\varphi\in H_\varphi$ is the image of $1\in M$ in $H_\varphi$, and $\pi_\varphi(x) m\Omega_\varphi = xm\Omega_\varphi$ for all $x,m\in M$.
\begin{enumerate}
\item
Show that $\pi_\varphi$ is normal.
\item
Deduce that if $\varphi$ is faithful, then $M\cong \pi_\varphi(M) \subset B(H_\varphi)$ is a von Neumann algebra acting on $H_\varphi$.
\end{enumerate}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $\Phi: M \to N$ is a unital $*$-algebra homomorphism between von Neumann algebras.
\begin{enumerate}
\item
Prove that the following conditions imply $\Phi$ is normal:
\begin{enumerate}
\item
$\Phi$ is SOT-continuous on the unit ball of $M$.
\item
$\Phi$ is WOT-continuous on the unit ball of $M$.
\item
Suppose $N=N''\subseteq B(H)$.
For a dense subspace $D\subseteq H$, $m\mapsto\langle \Phi(m)\eta, \xi\rangle$ is WOT-continuous on $M$ for any $\eta, \xi \in D$.
\end{enumerate}
\item (optional) Which of the conditions above are equivalent to normality of $\Phi$?
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:JMJ}
Let $M$ be a finite von Neumann algebra with a faithful $\sigma$-WOT continuous tracial state.
Let $L^2M = L^2(M,\operatorname{tr})$ where $\Omega$ is the image of $1_M$ in $L^2M$.
Identify $M$ with its image in $B(L^2M)$ by part (3) of Problem \ref{prob:NormalHom}.
\begin{enumerate}
\item
Show that $J: M\Omega \to M\Omega$ by $a\Omega \mapsto a^*\Omega$ is a conjugate-linear isometry with dense range.
\item
Deduce $J$ has a unique extension to $L^2M$, still denoted $J$, which is a conjugate-linear unitary, i.e, $J^2 = 1$ and $\langle J\eta, J\xi\rangle = \langle \xi, \eta\rangle$ for all $\eta,\xi\in L^2M$.
\\
\emph{Hint: Look at $\eta,\xi$ in $M\Omega$.}
\item
Calculate $Ja^*J b\Omega$ for $a,b\in M$.
Deduce that $JMJ\subseteq M'$.
\item
Show $\langle Ja^*J b\Omega, c\Omega\rangle = \langle b\Omega, JaJc\Omega \rangle$ for all $a,b,c\in M$.
Deduce $(JaJ)^* = Ja^*J$.
\item
Show $\langle Jy\Omega, a\Omega\rangle = \langle y^*\Omega, a\Omega\rangle$ for all $a\in M$ and $y\in M'$.
Deduce $Jy\Omega = y^*\Omega$.
\item
Prove that for $y\in M'$, $(JyJ)^* = Jy^*J$.
\\
\emph{Hint: Try the same technique as in (4).}
\item
Show for all $a,b\in M$ and $x,y\in M'$,
$\langle xJyJ a\Omega, b\Omega \rangle = \langle JyJ x a\Omega, b\Omega\rangle$.
\item
Deduce that $M'\subseteq (JM'J)' = JMJ$, and thus $M' = JMJ$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:LGamma}
Let $\Gamma$ be a discrete group, and let $L\Gamma =\{\lambda_g\}'' \subset B(\ell^2\Gamma)$.
Consider the faithful $\sigma$-WOT continuous tracial state $\operatorname{tr}(x) = \langle x \delta_e, \delta_e\rangle$ on $L\Gamma$.
\begin{enumerate}
\item
Show that $u\delta_g = \lambda_g$ uniquely extends to a unitary $u\in B(\ell^2 \Gamma, L^2L\Gamma)$ such that for all $x\in L\Gamma$ and $\xi \in \ell^2\Gamma$, $L_x u\xi = u x \xi$ where $L_x \in B(L^2L\Gamma)$ is left multiplication by $x$, i.e., $L_x(y\Omega) = xy\Omega$.
\item
Deduce from Problem \ref{prob:JMJ} that $L\Gamma' = R\Gamma$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:AlternateLGamma}
Use Problem \ref{prob:LGamma} above to give the following alternative characterization of $L\Gamma$.
Let
$$\ell\Gamma= \set{x=(x_g)\in \ell^2\Gamma}{ x * y \in \ell^2\Gamma\text{ for all }y\in \ell^2\Gamma}$$
where $(x*y)_g = \sum_h x_h y_{h^{-1}g}$.
Define a unital $*$-algebra structure on $\ell\Gamma$ by
multiplication is convolution, the unit is $\delta_e$, the the indicator function at $e\in\Gamma$ ($\delta_e(g) = \delta_{g=e}$), and the involution $*$ on $\ell\Gamma$ is given on $x\in \ell\Gamma$ by $(x^*)_g := \overline{x_{g^{-1}}}$.
\begin{enumerate}
\item
Show that $\ell\Gamma$ is a well-defined unital $*$-algebra under the above operations.
\item
For $x\in \ell\Gamma$ define $T_x : \ell^2\Gamma \to \ell^2\Gamma$ by $T_x y = x*y$.
Prove $T_x \in B(\ell^2\Gamma)$.
\\
\emph{Hint:
Show that for all $x\in \ell\Gamma$ and $y,z\in \ell^2\Gamma$, $\langle T_x y, z\rangle = \langle y, T_{x^*}z\rangle$.
Then use the Closed Graph Theorem.
}
\item
Prove that for all $x\in \ell\Gamma$, $T_x \in L\Gamma$.
\\
\emph{Hint: Prove $T_x \in R\Gamma'$ and apply Problem \ref{prob:LGamma}.}
\item
Deduce that $x\mapsto T_x$ is a unital $*$-algebra isomorphism $\ell\Gamma \to L\Gamma$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[V.~Jones]
Suppose $M=M_2(\bbC)$ and $\varphi$ is a state.
Then $\varphi(x)=\tr(x\rho)$ for a unique density matrix $\rho\geq 0$ with $\tr(\rho)=1$.
Choosing a basis of eigenvectors for $\rho$, we may identify
$$
\rho =
\begin{pmatrix}
\frac{1}{1+\lambda}
\\
&
\frac{\lambda}{1+\lambda}
\end{pmatrix}
$$
for some $0\leq \lambda \leq 1$.
Observe that $\varphi$ is faithful if and only if $0<\lambda<1$ if and only if $\rho$ is invertible.
\begin{enumerate}
\item
Describe as best you can $L^2(M,\phi)$ in terms of $\lambda$.
\item
Show that the action of $M$ on $L^2(M,\phi)$ is faithful.
\item
From this point on, assume $0<\lambda<1$.
Consider
$S:L^2(M,\varphi)\to L^2(M,\varphi)$ by $x\Omega\mapsto x^*\Omega$.
Compute the polar decomposition $S=J\Delta^{1/2}$ where $\Delta=S^*S$.
\item
Show that $M'=JMJ=SMS$ on $L^2(M,\varphi)$.
\item
Show that for all $z\in\bbC$, $\Delta^z M \Delta^{-z}=M$.
\item
Deduce that we have a 1-parameter group of unitaries $t\mapsto \sigma_t:=\Delta^{it}$ for $t\in \bbR$ which preserve $M$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Repeat Problem \ref{prob:AlternateLGamma} for the crossed product von Neumann algebra $M\rtimes_\alpha \Gamma$ acting on $L^2M\otimes \ell^2\Gamma \cong L^2(\Gamma, L^2M)$ where $M$ is a finite von Neumann algebra with faithful normal tracial state $\tr$, $\Gamma$ is a discrete group, and $\alpha : \Gamma \to \operatorname{Aut}(M)$ is an action.
Here, we define
\begin{align*}
\ell^2(\Gamma, M) &= \set{x: \Gamma \to M}{\sum_g \|x_g\Omega\|^2_{L^2M} <\infty}
\\
\ell^2(\Gamma, L^2M) &= \set{\xi: \Gamma \to L^2M}{\sum_g \|\xi_g\|^2 <\infty}
\text{ and}
\\
M\,\text{\reflectbox{$\propto$}}_\alpha\, \Gamma &= \set{x =(x_g) \in \ell^2(\Gamma, M)}{ x*\xi \in \ell^2(\Gamma,L^2M) \text{ for all } \xi\in \ell^2(\Gamma, L^2M)}.
\end{align*}
Here, the convolution action is given by
$(x*\xi)_g = \sum_{h} x_h v_h\xi_{h^{-1}g}$
where $v_h \in U(L^2M)$ is the unitary implementing $\alpha_u \in \operatorname{Aut}(M)$.
Define an analogous unital $*$-algebra structure on $M\Gamma$ and find a unital $*$-algebra isomorphism $M\,\text{\reflectbox{$\propto$}}_\alpha\,\Gamma \to M\rtimes_\alpha \Gamma$.
\\
\emph{Hint:
Similar to $L\Gamma$, some people write elements of $M\rtimes_\alpha \Gamma$ as formal sums $\sum_g x_g u_g$ which does not converge in any operator topology.
Rather, $\sum_g x_g u_g (\Omega \otimes \delta_e)$ converges in $L^2M \otimes \ell^2 \Gamma$.
These formal sums can be algebraically manipulated to obtain a unital $*$-algebra structure using the covariance condition $u_g m u_g^* = \alpha_g(m)$ for all $g\in \Gamma$ and $m\in M$.
Thus
$$
\left(\sum_g x_g u_g\right)^* = \sum_g u_g x_g^* = \sum_g u_g x_g^* u_g^* u_g = \sum_g \alpha_g(x_g^*) u_g.
$$
Thus for $x=(x_g)\in M\,\text{\reflectbox{$\propto$}}_\alpha\,\Gamma$, we define $(x^*)_g = \alpha_g(x_g^*)$.
A similar algebraic manipulation gives the formula for multiplication, which is similar to convolution, but involves the action.
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Prove that a $*$-isomorphism between von Neumann algebras is automatically normal.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $(X,\mu)$ is a measure space
and $T: X\to X$ is a measurable bijection preserving the measure class of $\mu$.
Let $\alpha_T\in\operatorname{Aut}(L^\infty(X,\mu))$ by $(\alpha_Tf)(x)=f(T^{-1}x)$.
Is it always the case that
the condition $\mu(\set{x\in X}{Tx=x})=0$
is equivalent to the automorphism $\alpha_T$ being free?
If yes, give a proof, and if not, find a counterexample together with a mild condition under which it is true.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $\bbF_2 = \langle a, b\rangle$ be the free group on 2 generators.
\begin{enumerate}
\item
Show that $\bbF_2$ is ICC.
Deduce $L\bbF_2$ is a ${\rm II}_1$ factor.
\item
Show that the swap $a\leftrightarrow b$ extends to an automorphism $\sigma$ of $L\bbF_2$.
\item
Show that $\sigma$ is outer.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\mbox{}
\begin{enumerate}
\item
(Fell's Absorption Principle)
Suppose $\Gamma$ is a countable group and $(H,\pi)$ is a unitary representation on a separable Hilbert space.
Find a unitary $u\in B(\ell^2\Gamma \overline{\otimes} H)$ intertwining $\lambda \otimes \pi$ and $\lambda \otimes 1$, i.e., $u (\lambda_g \otimes \pi_g) = (\lambda_g \otimes 1)u$ for all $g\in \Gamma$.
\item
Consider the two definitions of $M\rtimes_\alpha \Gamma$
when $(M,\tr)$ is a tracial von Neumann algebra and $\tr\circ \alpha_g = \tr$ for all $g\in \Gamma$.
The first is the von Neumann algebra generated by the $\pi_m$ and $u_g$ on $\ell^2(\Gamma,L^2M)$ where
$$
(u_g\xi)(h):= \xi(g^{-1}h)
\qquad\qquad
(\pi_m\xi)(h)=\alpha_{h^{-1}}(m)\xi(h).
$$
The second is the von Neumann algebra generated by the $\pi_m$ and $u_g$ on $L^2M\otimes \ell^2\Gamma$ given by
$$
\pi_m(x\Omega\otimes \delta_h)= mx\Omega\otimes \delta_h
\qquad\qquad
u_g(x\Omega\otimes \delta_h) = \alpha_g(x)\Omega \otimes \delta_{gh}.
$$
Find a unitary isomorphism $\ell^2(\Gamma,L^2M) \to L^2M\otimes \ell^2\Gamma$ intertwining the two $M$-actions and $\Gamma$-actions.
Deduce the two definitions of $M\rtimes_\alpha \Gamma$ are equivalent.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Prove that irrational rotation on the circle (with Lebesgue/Haar measure) is free and ergodic.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:L^1M}
Let $M$ be a finite von Neumann algebra with a faithful normal tracial state.
\begin{enumerate}
\item
Show for all $x,y\in M$, $|\tr(xy)|\leq \|y\|\tr(|x|)$.
\item
Show for all $x\in M$, $\tr(|x|) = \sup\set{|\tr(xy)|}{y\in M \text{ with }\|y\|=1}$.
\item
Define $\|x\|_1 = \tr(|x|)$ on $M$.
Show that $\|\cdot\|_1$ is a norm on $M$.
\item
Define a map $\varphi: M \to M_*$ by $x\mapsto \varphi_x$ where $\varphi_x(y) = \tr(xy)$.
Show that $\varphi$ is a well-defined isometry from $(M, \|\cdot\|_1) \to M_*$ with dense range.
\item
Deduce that $L^1(M, \tr) := \overline{M}^{\|\cdot\|_1}$ is isometrically isomorphic to the predual $M_*$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:ConditionalExpectation1}
Continue the notation of Problem \ref{prob:L^1M}.
Let $N\subseteq M$ be a (unital) von Neumann subalgebra.
\begin{enumerate}
\item
Prove that the inclusion $N\to M$ extends to an isometric inclusion $i:L^1(N, \tr) \to L^1(M,\tr)$.
\item
Let $E: M \to N$ be the Banach adjoint of $i$ under the identification $M_* = L^1(M,\tr)$ and $N_* = L^1(N,\tr)$.
Show that $E$ is uniquely characterized by the equation
$$
\tr_M(xy)= \tr_N(E(x)y)
\qquad
x\in M, y\in N.
$$
\end{enumerate}
\emph{Note: $E$ is called the canonical trace-preserving \emph{conditional expectation} $M \to N$.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:ConditionalExpectation2}
Suppose $M$ is a finite von Neumann algebra with normal faithful tracial state $\tr$ and $N\subseteq M$ is a (unital) von Neumann subalgebra.
\begin{enumerate}
\item
Prove that the inclusion $N\to M$ extends to an isometric inclusion $L^2(N,\tr) \to L^2(M,\tr)$.
\item
Define $e_N \in B(L^2M, L^2N)$ be the orthogonal projection with range $L^2(N,\tr)= \overline{N\Omega}^{\|\cdot\|_2} \subset L^2(M,\tr)$.
Show that for all $x\in M$, $e_Nxe_N^* \subset B(L^2N)$ commutes with the right action of $N$, and thus defines an element in $N$ by Problem \ref{prob:JMJ}.
\\
\emph{Hint: Show the inclusion $e_N^* : L^2N \to L^2M$ commutes with the right $N$ action, and deduce $e_N$ commutes with the right $N$ action.}
\item
For $x\in M$, define $E(x) = e_N xe_N^*$.
Show that $E(x)$ is uniquely characterized by the equation
$$
\tr_M(xy)= \tr_N(E(x)y)
\qquad
x\in M, y\in N.
$$
\end{enumerate}
\emph{Note: $E$ is called the canonical trace-preserving \emph{conditional expectation} $M \to N$.
Part (3) implies this definition agrees with that from Problem \ref{prob:ConditionalExpectation1}.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Continue the notation of Problem \ref{prob:ConditionalExpectation2}.
\begin{enumerate}
\item
Deduce that $E$ is normal.
\item
Deduce $E(1) = 1$ and $E$ is $N-N$ bilinear, i.e., for all $x\in M$ and $y,z\in N$, $E(yxz) = yE(x)z$.
\item
Deduce that $E(x^*) = E(x)^*$.
\item
Show that $E$ is completely positive, which was defined in Problem \ref{prob:CompletelyPositive}.
\\
\emph{Hint: Use the characterization $E(x) = e_Nxe_N^*$ from (5) of Problem \ref{prob:ConditionalExpectation2}.}
\item
Show that $E(x)^*E(x) \leq E(x^*x)$ for all $x\in M$.
\\
\emph{Hint: Use the characterization $E(x) = e_Nxe_N^*$ from (5) of Problem \ref{prob:ConditionalExpectation2}.
Show that $e_N^*e_N$ is an orthogonal projection.}
\item
Show that $E$ is faithful: $E(x^*x) = 0$ implies $x^*x=0$.
\\
\emph{Hint: Prove this by looking at the vector states $\omega_{n\Omega}$ for $n\in N$.}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $M$ is a finite von Neumann algebra with faithful normal tracial state $\tr$.
Suppose further that there is an increasing sequence of von Neumann subalgebras $M_1 \subset M_2 \subset \cdots M$ such that $(\bigcup M_n)'' = M$ (considered as acting on $L^2M$).
Let $E_n : M \to M_n$ be the canonical trace-preserving conditional expectation from Problem \ref{prob:ConditionalExpectation2}.
\begin{enumerate}
\item
Prove that the $\|\cdot\|_2$-topology agrees with the SOT on the unit ball of $M$.
That is, prove that $x_n \to x$ SOT if and only if $\|x_n\Omega -x\Omega\|_2 \to 0$.
\item
Prove that for all $x\in M$, $\|E_n(x)\Omega -x\Omega\|_2 \to 0$ as $n\to \infty$.
\item
Deduce that $E_n(x) \to x$ SOT as $n\to \infty$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:Amenable}
Suppose $\Gamma$ is a countable group, and let
$\Prob(\Gamma) = \set{ \mu \in \ell^1\Gamma}{\mu\geq 0\text{ and }\sum_g \mu(g)=1}$.
\begin{enumerate}
\item
Prove that $\Prob(\Gamma)$ is weak* dense in the state space of $\ell^\infty\Gamma$.
\item
Let $F\subset \Gamma$ be finite, and consider $\bigoplus_{g\in F} \ell^1\Gamma$ with the (product) weak topology.
Let $K$ be the weak closure of $\set{\bigoplus_{g\in F} g\cdot \mu - \mu }{\mu \in \Prob(\Gamma)}\subset \bigoplus_{g\in F} \ell^1\Gamma$.
Prove $K$ is convex and norm closed in $\bigoplus_{g\in F} \ell^1\Gamma$.
\item
Now assume $\Gamma$ is amenable, i.e., there is a left $\Gamma$-invariant state on $\ell^\infty \Gamma$.
Prove that $0\in K$.
Deduce that $\Gamma$ has an approximately invariant mean.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $\Gamma$ is a countable group, and let $\Prob(\Gamma)$ be as in Problem \ref{prob:Amenable}.
%For $r\in [0,1]$, let $\chi_r = \chi_{(r,1]}$ be the characteristic function.
\begin{enumerate}
\item
Prove that if $a,b\in [0,1]$, then
$$
|a-b|
=
\int_0^1 |\chi_{(r,1]}(a) - \chi_{(r,1]}(b)| \, dr.
$$
\item
Deduce that for $\mu \in \Prob(\Gamma)$ and $h\in \Gamma$,
$$
\|h\cdot \mu - \mu\|_{\ell^1\Gamma}
=
\int_0^1 \sum_{g\in \Gamma} |\chi_{(r,1]}(\mu(h^{-1}g)) - \chi_{(r,1]}(\mu(g))| \, dr.
$$
\item
For $r\in [0,1]$ and $\mu \in \Prob(\Gamma)$, let $E(\mu,r) = \set{g\in \Gamma}{\mu(g)>r}$.
Show that for all $h\in \Gamma$, $hE(\mu,r) = \set{g\in \Gamma}{(h\cdot \mu)(g)>r}$.
\item
Calculate $\int_0^1 |E(\mu, r)| \, dr$.
\item
Show that for $r\in [0,1]$, $\mu \in \Prob(\Gamma)$, and $h\in \Gamma$,
$$
|hE(\mu,r) \triangle E(\mu,r)|
=
\sum_{g\in \Gamma} |\chi_{(r,1]}(\mu(h^{-1}g)) - \chi_{(r,1]}(\mu(g))|.
$$
Deduce that $\|h\cdot \mu - \mu\|_1 = \int_0^1 |h E(\mu,r) \triangle E(\mu,r)|\, dr$.
\item
Suppose now that $\Gamma$ has an approximate invariant mean, so that for every finite subset $F \subset \Gamma$ and $\varepsilon>0$, there is a $\mu \in \Prob(\Gamma)$ such that
$$
\sum_{h\in F} \|h \cdot \mu - \mu\|_1 < \varepsilon.
$$
Show that for the $\mu$ corresponding to this $F$ and $\varepsilon$,
$$
\int_0^1 \sum_{h\in F} |h E(\mu,r) \triangle E(\mu,r)|\, dr <\varepsilon \int_0^1 |E(\mu, r)| \, dr.
$$
Deduce there is an $r\in [0,1]$ such that $|h E(\mu, r) \triangle E(\mu, r)| < \varepsilon |E(\mu, r)|$ for all $h\in F$.
\item
Use (6) above to construct a F{\o}lner sequence for $\Gamma$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Recall that an \emph{ultrafilter} $\omega$ on a set $X$ is a nonempty collection of subsets of $X$ such that:
\begin{itemize}
\item
$\emptyset \notin \omega$,
\item
If $A\subseteq B\subseteq X$ and $A\in \omega$, then $B\in \omega$,
\item
If $A,B\in \omega$, then $A\cap B \in \omega$, and
\item
For all $A\subset X$, either $A\in \omega$ or $X\setminus A \in \omega$ (but not both!).
\end{itemize}
\begin{enumerate}
\item
Find a bijection from the set of ultrafilters on $\bbN$ to $\beta\bbN$, the Stone-Cech compactification of $\bbN$.
\item
Let $\omega$ be an ultrafilter on $\bbN$.
Let $X$ be a compact Hausdorff space and $f: \bbN \to X$.
We say
\begin{itemize}
\item $x = \lim_{n\to \omega} f(n)$ if for every open neighborhood $U$ of $x$, $f^{-1}(U) \in \omega$.
\end{itemize}
Prove that $\lim_{n\to \omega} f(n)$ always exists for any function $f: \bbN \to X$.
\item
An ultrafilter on $\bbN$ is called \emph{principal} if it contains a finite set.
Show that every principal ultrafilter on $\bbN$ contains a unique singleton set, and that any two principal ultrafilters containing the same singleton set are necessarily equal.
Thus we may identify the set of principal ultrafilters on $\bbN$ with $\bbN \subset \beta \bbN$.
\item
Determine $\lim_{n\to \omega} f(n)$ for $f: \bbN \to X$ as in (2) when $\omega$ is principal.
\item
An ultrafilter on $\bbN$ is called \emph{free} or \emph{non-principal} if it does not contain a finite set.
Let $\omega$ be a free ultrafilter on $\bbN$.
Suppose $\Gamma = \bigcup \Gamma_n$ is a locally finite group and $m_n$ is the uniform probability (Haar) measure on $\Gamma_n$.
Define $m: 2^\Gamma \to [0,1]$ by $m(A) = \lim_{n\to \omega} m_n(A\cap \Gamma_n)$.
Prove that $m$ is a left $\Gamma$-invariant finitely additive probability measure on $\Gamma$, i.e., $\Gamma$ is amenable.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:ExistsUniqueFixedPoint}
Let $X$ be a uniformly convex Banach space and $B\subset X$ a bounded set.
Prove that the function $f: X \to [0,\infty)$ given by $f(x) = \sup_{b\in B} \|b - x\|_X$ achieves its minimum at a unique point of $X$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $\Gamma$ be a countable discrete group.
Show that an affine action $\alpha = (\pi,\beta): \Gamma \to \operatorname{Aff}(H)$ ($\alpha_g \xi := \pi_g \xi + \beta(g)$ for $\pi_g\in U(H)$ and $\beta(g)\in H$ such that $\alpha_g \circ \alpha_h = \alpha_{gh}$ for all $g,h\in \Gamma$) is proper if and only if the cocycle part $\beta: \Gamma \to H$ is proper ($g\mapsto \|\beta(g)\|$ is a proper map).
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Recall that the \emph{Schur product} of two matrices $a,b\in M_n(\bbC)$ is given by the entry-wise product: $(a*b)_{i,j} : = a_{i,j}b_{i,j}$.
\begin{enumerate}
\item
Prove that if $a,b\geq 0$, then $a*b \geq 0$.
\item
Suppose that $p \in \bbR[z]$ is a polynomial whose coefficients are all non-negative.
Prove that if $a\geq 0$, then $p[a]\geq 0$, where $p[a]_{i,j} := p(a_{i,j})$ for $a\in M_n(\bbC)$.
\\
\emph{Note: Here we use the notation $p[a]$ to not overload the functional calculus notation.}
\item
Suppose that $f$ is an entire function whose Taylor expansion at $0$ has only non-negative real coefficients.
Prove that is $a\geq 0$, then $f[a]\geq 0$, where again $f[a]_{i,j} := f(a_{i,j})$ for $a\in M_n(\bbC)$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $A$ be a unital C*-algebra.
\begin{enumerate}
\item
Prove that a map $\Phi : A \to M_n(\bbC)$ is completely positive if and only if the map $\varphi:M_n(A) \to \bbC$ given by $(a_{i,j}) \mapsto \sum_{i,j}^n\Phi(a_{i,j})_{i,j}$ is positive.
\\
\emph{Hint: for one direction, note that $\varphi(a) = \vec{e}^* \Phi(a)\vec{e}$ where $\vec{e}\in \bbC^{n^2}$ is the vector $(e_1, e_2,\dots, e_n)$ where $e_i \in \bbC^n$ is the $i$-th standard basis vector.
For the other direction, use GNS with respect to $\varphi$, and consider $V: \bbC^n \to L^2(M_n(A), \varphi)$ given by $Ve_i = \pi_{\varphi}(E_{ij}) \Omega_{\varphi}$ where $(E_{ij})$ is a system of matrix units in $M_n(\bbC) \subseteq M_n(A)$.
Then use Stinespring.}
\item
Let $S\subset A$ be an operator subsystem, and let $\psi : S \to \bbC$ be a positive linear functional.
Prove $\|\psi\|=\psi(1)$.
Deduce that any norm-preserving (Hahn-Banach) extension of $\psi$ to $A$ is also positive.
\item
Let $S\subset A$ be an operator subsystem, and let $\Phi : S\to M_n(\bbC)$ be a (unital) completely positive map.
Show that $\Phi$ extends to a (unital) completely positive map $A\to M_n(\bbC)$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $\Gamma$ is a countable discrete group, and suppose $\varphi: L\Gamma \to L\Gamma$ is a normal completely positive map.
Prove that $f: \Gamma \to \bbC$ given by $f(g):=\tr_{L\Gamma}(\varphi(\lambda_g)\lambda_g^*)$ is a positive definite function.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Prove that the following are equivalent for a finite von Neumann algebra $(M,\tr)\subset B(H)$ with faithful normalized tracial state.
\begin{enumerate}
\item
$M$ is amenable, i.e., there is a conditional expectation $E: B(H) \to M$.
\item
There is a sequence $(\varphi_n: M \to M)$ of (normal) trace-preserving completely positive maps such that $\varphi_n \to \operatorname{id}$ pointwise in $\|\cdot\|_M$, and for all $n\in \bbN$, the induced map $\widehat{\varphi}_n \in B(L^2M)$ given by $m\Omega \mapsto \varphi_n(m)\Omega$ is finite rank.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose that $\Gamma$ is a countable discrete group such that every cocycle is inner.
Suppose $(H,\pi)$ is a unitary representation and $(\xi_n)\subset H$ is a sequence of unit vectors such that $\|\pi_g \xi_n -\xi_n\| \to 0$ as $n\to \infty$ for all $g\in \Gamma$.
Follow the steps below to find a non-zero $\Gamma$-invariant vector in $H$.
(We may assume that no $\xi_n$ is fixed by $\Gamma$.)
\begin{enumerate}
\item
Enumerate $\Gamma = \{g_1,g_2,\dots\}$.
Explain why you can pass to a subsequence of $(\xi_n)$ to assume that for all $n\in \bbN$, $\|\pi_{g_i} \xi_n - \xi_n\| < 4^{-n}$ for all $1\leq i \leq n$.
\item
For $n\in \bbN$, consider the inner cocycles $\beta_n(g):= \xi_n - \pi_g \xi_n$.
Let $(K, \sigma) = \bigoplus_{n\in\bbN} (H,\pi)$.
Define $\beta : \Gamma \to K$ by $\beta(g)_n := 2^n\beta_n(g)$.
Prove that $\beta(g)\in H$ is well-defined for every $g\in \Gamma$.
Then show that $\beta$ is a cocycle for $(K,\sigma)$.
\item
Deduce $\beta$ is inner and thus bounded.
Thus there is a $\kappa \in K\setminus \{0\}$ such that $\beta(g) = \kappa - \sigma_g \kappa$ for all $g\in \Gamma$.
\item
Prove that $\|\beta_n(g)\| \to 0$ uniformly for $g\in \Gamma$.
That is, show that for all $\varepsilon>0$, there is an $N\in \bbN$ such that $n>N$ implies $\|\beta_n(g)\|<\varepsilon$ for all $g\in\Gamma$.
\item
Fix $N\in \bbN$ such that $\|\beta_N(g)\|= \|\xi_N - \pi_g \xi_N\| <1$ for all $g\in \Gamma$.
Show there is a $\xi_0 \in H\setminus \{0\}$ such that $\pi_g \xi_0 = \xi_0$ for all $g\in \Gamma$.
\\
\emph{Hint: Look at $\set{\pi_g \xi_N}{g\in\Gamma}\subset (H)_1$ and apply Problem \ref{prob:ExistsUniqueFixedPoint}.}
\item
(optional)
Use a similar trick to finish the proof of $(1)\Rightarrow (2)$ from the same theorem from class.
\end{enumerate}
\end{prob}
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\begin{prob}[optional]
As best as you can, edit the equivalent definitions I gave in class for property (T) for a countable discrete group $\Gamma$ to be relative to a subgroup $\Lambda \leq \Gamma$.
Then prove all the equivalences.
\end{prob}
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\begin{prob}
Suppose $\Gamma \curvearrowright (X, \mu)$ is a free p.m.p.~action and $\cR = \set{(x,gx)}{x\in X, g\in \Gamma}$ is the corresponding countable p.m.p.~equivalence relation.
Follow the steps below to show $L^\infty(X, \mu)\rtimes \Gamma \cong L\cR$.
\begin{enumerate}
\item
Prove that $\theta: (x,g)\mapsto (x,g^{-1}x)$ induces a unitary operator $v\in B(L^2\cR , L^2(X\times \Gamma, \mu\times \gamma))$ where $\gamma$ is counting measure on $\Gamma$.
\item
Deduce that $\theta$ is a p.m.p.~isomorphism $(X\times \Gamma, \mu\times \gamma) \to (\cR, \nu)$.
\item
Show that $v^* M_f v = \lambda(f)$ for all $f\in L^\infty(X,\mu)$.
Here, $(M_f\xi)(x,g) = f(x)\xi(x,g)$ for $\xi \in L^2(X\times \Gamma, \mu\times \gamma)$.
\item
Show that $v^* u_g v = L_{\varphi_g}$ where $\varphi_g \in [\cR]$ is the isomorphism $x\mapsto g\cdot x$.
Here, $(u_g\xi)(x,h) = \xi(g^{-1}x, g^{-1}h)$ for all $\xi \in L^2(X\times \Gamma, \mu\times \gamma)\cong L^2(X,\mu)\otimes \ell^2\Gamma$.
\item
Deduce that $v^* (L^\infty(X,\mu) \rtimes \Gamma) v \subset L\cR$.
\item
Show that conjugation by $v$ takes the commutant of $L^\infty(X,\mu) \rtimes \Gamma$ into $R\cR$.
\\
\emph{Hint: Show that right multiplication by $L^\infty(X, \mu)$ and the right action of $u_g$ are both taken into $R\cR$.}
\item
Deduce that $v^* (L^\infty(X,\mu) \rtimes \Gamma) v = L\cR$.
\end{enumerate}
\end{prob}
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\begin{prob}
Let $\cR$ be a countable p.m.p.~equivalence relation on $(X,\mu)$.
Let $A = L^\infty(X,\mu) \subset L\cR$.
Prove that the von Neumann subalgebra of $B(L^2(\cR,\nu))$ generated by $A\cup JAJ$ is the von Neumann algebra of multiplication operators by elements of $L^\infty(\cR, \nu)$.
\end{prob}
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%\begin{prob}
%\begin{enumerate}
%\item
%Suppose $\cR_i$ is a p.m.p.~equivalence relation on $(X_i, \mu_i)$ for $i=1,2$, and $\theta: (X_1, \mu_1) \to (X_2, \mu_2)$ is a p.m.p.~isomorphism.
%Let $v\in B(L^2(X_1, \mu_1) , L^2(X_2, \mu_2))$ be the associated unitary isomorphism given by $f\mapsto f\circ \theta^{-1}$.
%Show that the following are equivalent:
%\begin{enumerate}
%\item
%Up to null sets, $(\theta \times \theta) (\cR_1) = \cR_2$.
%\item
%Conjugation by $v$ provides a spatial unital $*$-algebra isomorphism $L(\cR_1) \cong L(\cR_2)$.
%\end{enumerate}
%\item
%Suppose $\Gamma_i \curvearrowright (X_i, \mu_i)$ is a p.m.p.~action for $i=1,2$, and $\theta: (X_1, \mu_1) \to (X_2, \mu_2)$ is a p.m.p.~isomorphism.
%Deduce that the following are equivalent:
%\begin{enumerate}
%\item
%$\theta$ is an orbit equivalence.
%\item
%$L^\infty(X_1, \mu_1)\rtimes \Gamma_1 \cong L^\infty(X_2, \mu_2) \rtimes \Gamma_2$.
%\end{enumerate}
%\end{enumerate}
%\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $M$ be a von Neumann algebra.
A \emph{weight} on $M$ is a function $\varphi: M_+\to[0,\infty]$ such that
%\begin{itemize}
%\item(weight)
for all $r\in [0,\infty)$ and $x,y\in B(H)_+$, $\varphi(rx+y) = r\varphi(x)+\varphi(y)$, with the convention that for $s\in [0,\infty)$,
$$\infty \cdot s =
\begin{cases}
\infty &\text{if }s>0
\\
0 &\text{if }s=0.
\end{cases}$$
%\item (normal)
%$x_\lambda \nearrow x$ (as in Problem \ref{prob:IncreasingNet}) implies $\varphi(x_\lambda) \nearrow \varphi(x)$.
%\end{itemize}
Define
\begin{align*}
\mathfrak{p}_\varphi
&=
\set{x\in M}{\varphi(x)<\infty}
\\
\mathfrak{n}_\varphi
&=
\set{x\in M}{x^*x\in \mathfrak{p}_\varphi}
\\
\mathfrak{m}_\varphi
&=
\mathfrak{n}_\varphi^*\mathfrak{n}_\varphi
=
\set{\sum_{i=1}^n x_i^*y_i}{x_i,y_i \in \mathfrak{n}_\varphi\text{ for all }i=1,\dots, n}.
\end{align*}
\begin{enumerate}
\item
Prove that
\begin{enumerate}
\item
$\mathfrak{p}_\varphi$ is a hereditary subcone of $M_+$, i.e.,
\begin{itemize}
\item (subcone) $r\geq 0$ and $x,y\in \mathfrak{p}_\varphi$ implies $rx+y \in \mathfrak{p}_\varphi$
\item (hereditary) $0\leq x\leq y$ and $y\in \mathfrak{p}_\varphi$ implies $x\in \mathfrak{p}_\varphi$.
\end{itemize}
\item
$\mathfrak{n}_\varphi$ is a left ideal of $M$.
\\
\emph{Hint: Prove that for all $x,y\in M$, $(x\pm y)^*(x\pm y) \leq 2(x^*x+y^*y)$.}
\item
$\mathfrak{m}_\varphi$ is algebraically spanned by $\mathfrak{p}_\varphi$.
\\
\emph{Hint: Use polarization.}
\item
$\mathfrak{m}_\varphi\cap M_+ = \mathfrak{p}_\varphi$.
\item
$\mathfrak{m}_\varphi$ is a hereditary $*$-subalgebra of $M$ (hereditary is defined the same way as above).
\end{enumerate}
\item
When $M=B(H)$ and $\varphi = \operatorname{Tr}$, show $\mathfrak{m}_{\operatorname{Tr}} = \cL^1(H)$ and $\mathfrak{n}_{\operatorname{Tr}} = \cL^2(H)$.
\end{enumerate}
\end{prob}
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\end{document}