We will first consider completely positive maps and completely bounded maps between C*-algebras and motivate why one would want to consider such morphisms. We will then define abstract operator systems, abstract operator spaces, and compare them with their concrete counterparts.

• 1. Verify that every concrete operator system is an abstract operator system.
• 2. Let $\mathcal{A}$ be a unital C*-algebra with unit 1. If $a,b\in \mathcal{A}$ prove that \[ \left(\begin{array}{cc} 1 & a \\ a^* & b \end{array}\right) \] is positive if and only if $b-a^*a$ is positive.
• 3. Let $\mathcal{X}$ and $\mathcal{Y}$ be operator spaces and let $u\colon \mathcal{X}\to \mathcal{Y}$ be a linear map. If $u_n\colon M_n(\mathcal{X})\to M_n(\mathcal{Y})$ denotes the $n$th amplification, prove that $u_n$ is an $M_n(\mathbb{C})$-bimodule map.
• 4. (The Paulsen System) Let $\mathcal{A}$ and $\mathcal{B}$ be unital C*-algebras with unit $1$ and let $\mathcal{X}\subset \mathcal{A}$ be an operator space. Consider the following operator system $\mathcal{V}_{\mathcal{X}}\subset M_2(\mathcal{A})$: \[ \mathcal{V}_{\mathcal{X}}:= \left\{ \left(\begin{array}{cc} \lambda 1 & x \\ y^* & \mu 1 \end{array}\right) \colon \lambda,\mu\in \mathbb{C},\ x,y\in \mathcal{X} \right\}. \] If $u\colon \mathcal{X}\to \mathcal{B}$ is a completely contractive map into the C*-algebra $\mathcal{B}$, prove that the map defined by \[ \varphi_{u}\colon \mathcal{V}_{\mathcal{X}} \to M_2(\mathcal{B})\\ \qquad\left(\begin{array}{cc} \lambda1 & x \\ y^* & \mu 1 \end{array}\right) \mapsto \left(\begin{array}{cc} \lambda 1 & u(x) \\ u(y)^* & \mu 1 \end{array}\right) \] is completely positive.

Here we will (gently) dive into arguably the most famous problem in operator algebras, Kirchberg's conjecture. In his seminal work in 1993, Eberhard Kirchberg proved that a remark made by Alain Connes was equivalent to a "universal" group C*-algebra having a property known as Lance's Weak Expectation Property. We will begin by discussing the weak expectation property for C*-algebras. Afterwards we shall take a stroll along the road from Connes '76 to MIP*=RE '20.

I will motivate the definition of a non commutative probability space, an example which in fact includes all “standard” probability spaces (in much the same way that C*-algebras include algebras of continuous functions on compact Hausdorff spaces). I will then define the notion of “distribution,” which will simplify to the notion of distribution in standard probability in the case of a normal element. From there, I will cover several important examples of distributions in free probability, in particular the semicircular law. Finally, I will define the notion of free independence, and will present several examples including examples coming from free products of groups.

• 1. Let $(\mathcal{A},\varphi)$ be a $*$-probability space. In this exercise you will show that classical and free independence do not mix well.
  • (a) For $a_1,a_2,b\in \mathcal{A}$, suppose $\{a_1,a_2\}$ is $*$-free from $\{b\}$. Show that $\varphi(a_1ba_2)= \varphi(a_1 a_2)\varphi(b)$.
  • (b) Show that if $a,b\in \mathcal{A}$ are $*$-free, then \[ \varphi(a^*b^*ab)= |\varphi(a)|^2\varphi(b^*b) + \varphi(a^*a)|\varphi(b)|^2 - |\varphi(a)|^2 |\varphi(b)|^2. \]
  • (c) Let $a$ and $b$ be as in the previous part, and suppose that in addition to being $*$-free they also commute with one another. Use the previous part as well as a new expression for $\varphi(a^*b^*ab)$ to show that \[ \varphi(a^*a)\varphi(b^*b) - |\varphi(a)|^2\varphi(b^*b) - \varphi(a^*a)|\varphi(b)|^2 + |\varphi(a)|^2 |\varphi(b)|^2 =0. \] Use this equation to show that \[ \varphi([a- \varphi(a)]^*[a - \varphi(a)])\varphi([b-\varphi(b)]^*[b- \varphi(b)])=0. \] Deduce that if $\varphi$ is faithful and $a$ and $b$ are $*$-free and commute, then at least one of $a$ or $b$ is a scalar multiple of the identity.
• 2. Let $(\mathcal{A},\varphi)$ and $(\mathcal{B},\psi)$ be (C* or W*) probability spaces generated by $a=(a_1,\ldots, a_n)$ and $b=(b_1,\ldots, b_n)$, respectively. Suppose $\varphi$ and $\psi$ are faithful and that the laws $\mu_a$ and $\mu_b$ are equal. Show that $\mathcal{A}$ and $\mathcal{B}$ are isomorphic as (C* or W*) algebras.
  [Hint: think spatially. There should be a natural unitary between the GNS Hilbert spaces for $\varphi$ and $\psi$. What does the unitary do in regards to the GNS representations of $\mathcal{A}$ and $\mathcal{B}$?]
  (This exercise shows that the joint law of a tuple of elements uniquely determines the C*-algebra or von Neumann algebra said tuple generates.)
• 3. Let $x_1,\ldots, x_n$ be a free family of self-adjoint semicircular elements with variance $1$ in a C*-probability space $(\mathcal{A},\varphi)$. Show that \[ \frac{x_1+\cdots +x_n}{\sqrt{n}} \] is semicircular with variance $1$.
  (This exercise is meant to demonstrate the stability of the semicircular law under addition of freely independent elements. This is in analogy to the stability of the Gaussian law under sums of classically independent elements.)
• 4. Let $(\mathcal{A},\varphi)$ be a C*-probability space. Suppose $u\in A$ is a Haar unitary and $a\in \mathcal{A}$ is $*$-free from $u$. Show that the set \[ \{ u^k a u^{-k}\colon k\in \mathbb{Z}\} \] is a $*$-free family.
• 5. In this exercise, we examine the distribution of a sum of two $*$-free projections with first moment $\frac12$. Fix a C*-probability space $(\mathcal{A},\varphi)$.
  • (a) Let $u$ be a Haar unitary in $\mathcal{U}$. Show that the distribution of $u+u^*$ is the arcsine law: \[ \frac{1}{\pi\sqrt{4-t^2}}\ dt. \] [Hint: To start, argue why $\varphi((u+u^*)^n) = \frac{1}{2\pi} \int_0^{2\pi} (2\cos(t))^n\ dt = \frac{1}{\pi}\int_0^\pi (2\cos(t))^n\ dt$. From there, use a substitution.]
  • (b) Let $v$ and $w$ be self-adjoint centered unitary elements which are $*$-free from each other. Show that $vw$ is a Haar unitary in $\mathcal{A}$.
  • (c) Show that $wvw$ is $*$-free from $v$.
  • (d) Show that $vw+wv$ and $wvw+v$ have the same moments and hence the same law (since both elements are self-adjoint). Deduce that the law of a sum of two $*$-free centered self-adjoint unitary elements is also the arcsine law.
    [Hint: First show that $(vw+wv)^{2k} = (wvw+v)^{2k}$ for each positive $k$. Then, argue that every odd moment of $wvw+v$ is zero.]
  • (e) Now suppose $p$ and $q$ are two $*$-free projections satisfying $\varphi(p)=\varphi(q) = \frac{1}{2}$. Use the result of the previous part to find the distribution of $p+q$.
  (The above exercise shows a theme in free probabity: combining free elements often "kills off" atoms from the law. In this exercise, two free elements with atomic distributions were added together to produce a distribution with no atoms anywhere. Can this happen in the classically independent case?)
• 6. In this exercise, you may assume the following fact without proof: if $p$ and $q$ are $*$-free projections in a W*-probability space with $\varphi(p)\leq \varphi(q)$, then there is a partial isometry $v\in W^*(p,q)$ satisfying $v^*v=p$ and $vv^*\leq q$.
  Suppose $(\mathcal{A},\varphi)$ contains $M_2(\mathbb{C})$ as a unital von Neumann subalgebra so that for some $\lambda\in (0,\frac12)$ one has \[ \varphi\left( \begin{array}{cc} a & b \\ c & d \end{array}\right) = \text{Tr}\left( \left( \begin{array}{cc} a & b \\ c & d \end{array}\right)\left( \begin{array}{cc} \lambda & 0 \\ 0 & 1-\lambda \end{array}\right)\right) \qquad \left( \begin{array}{cc} a & b \\ c & d \end{array}\right)\in M_2(\mathbb{C}). \] Show that if $\mathcal{A}$ contains a projection $p$ with $\varphi(p)=\frac12$ that is $*$-free from this copy of $M_2(\mathbb{C})$, then $\mathcal{A}$ contains a non-unitary isometry and hence does not have a faithful tracial state.

I will present some Voiculescu’s landmark theorems that, loosely speaking, describe freeness as a random-matrix limit of classical independence. I will discuss numerous applications of these theorems to the structure of free group factors, and use these applications to motivate the definition of “interpolated” free group factors $L(F_{t})$ for any real $t > 1$.

We will state the definition of amenability for groups and build intuition by looking at examples and non examples. Then we will define amenability for finite von Neumann algebras, and prove that a group is amenable only if its group von Neumann algebra is amenable.

• 1. Let $N$ be an amenable von Neumann algebra. Show that $M_d(N)$ is amenable for all $d\in \mathbb{N}$.
• 2. In this exercise, you will prove some interesting characterizations of amenability. Let $\Gamma$ be a countable group. Denote by $\lambda\colon \Gamma \to B(\ell^2(\Gamma))$ the left regular representation. Show that $(i)\Longrightarrow (ii) \Longleftrightarrow (iii)$ in the following:
  • $(i)$ For any unitary representation $\pi\colon \Gamma \to U(\mathcal{H})$ and any $\xi\in \ell^1(\Gamma)$ we have $\|\pi(\xi)\| \leq \| \lambda(\xi)\|$.
  • $(ii)$ For any finite set $F\subset \Gamma$ we have $\| \sum_{s\in F} \lambda(s) \| = |F|$.
  • $(iii)$ For any $\epsilon>0$ and finite set $F\subset \Gamma$ there exists a unit vector $\xi\in \ell^2(\Gamma)$ such that $\sum_{s\in F}\|\lambda(s)\xi - \xi\|^2 <\epsilon$.
  Challenging problem: prove that all of the above are equivalent. Further, show that $(i)$ is equivalent to the fact that $C_r^*(\Gamma)\cong C^*(\Gamma)$.
• 3. Show that if $G$ is amenable, then there is a conjugation invariant state on $\ell^\infty(G)$. Note that $G$ acts by conjugation on this C*-algebra in the natural way.
• 4. If $N$ is a finite von Neumann algebra with a faithful normal tracial state $\tau$, we say $N$ has property $\Gamma$ if for any $\epsilon>0$ and finite set $F\subset N$ there exists a unitary $u\in N$ with $\tau(u)=0$ and $\|\sum_{x\in F} [x,u]\|_2 <\epsilon$. This means $N$ almost has a non-trivial center. Show that $R$ has property $\Gamma$.
• 5. If $N\subset B(\mathcal{H})$ and $M\subset B(\mathcal{K})$ are von Neumann algebras, their von Neumann algebra tensor product $N\bar{\otimes}M$ is defined as $(M\odot N)'' \subset B(\mathcal{H}\otimes \mathcal{K})$ (equivalently, the SOT or WOT closure of the algebraic tensor product). Show that $R\bar{\otimes} L(\mathbb{F}_2)$ has property $\Gamma$. More generally, if $N$ has property $\Gamma$ show that $N\bar\otimes M$ has property $\Gamma$ for any finite von Neumann algebra $M$.
  Very, very, very, very challenging problem: show that if $N\bar\otimes M$ has property $\Gamma$, then either $N$ or $M$ has property $\Gamma$ (see Connes 1976 for a hint).

We will introduce inner amenability for groups, and build intuition by looking at examples and non examples. Then we will define the notions of property Gamma and McDuff for von Neumann algebras, and provide examples and non examples. We will briefly say why these notions are of fundamental importance in the program of classifying separable $\mathrm{II}_1$ factors.

In the first lecture, we begin with a review of classical objects from which we like to build C*-algebras: directed graphs, {0,1}-matrices, and Markov shift spaces, and we note the relationship between these objects. Next, we will define Cuntz-Krieger algebras, do a few examples, and state some notable properties and theorems of Cuntz-Krieger algebras (for finite {0,1}-matrices). Last, we will explore the natural relationship between Cuntz-Krieger algebras and graph C*-algebras.

Let $E=(E^0,E^1,r,s)$ be a row-finite directed graph with no sinks and no sources.

• 1. Let $A$ denote the edge matrix of $E$, i.e. for all $e,f\in E^1$ set $A(e,f)=1$ if $r(f)=s(e)$, and $A(e,f)=0$ otherwise.
  • (a) Given a Cuntz–Krieger $E$-family $(S,P)$, show that $\{S_e\colon e\in E^1\}$ forms a Cuntz–Krieger $A$-family.
  • (b) How would you go about showing that $C^*(E)$ is isomorphic to $\mathcal{O}_A$?
• 2. (Example 8.3, 8.7 - Raeburn) For a C*-algebra $\mathcal{A}$, an $\mathcal{A}$-correspondence is a right $A$-module $X$ equipped with:
  • (i) a representation $\phi\colon \mathcal{A}\to \mathcal{L}(X)$;
  • (ii) a bilinear map $\langle\cdot,\cdot\rangle\colon X\times X\to \mathcal{A}$ that satisfies \[ \langle x, y\cdot a\rangle = \langle x,y\rangle a,\qquad \langle x\cdot a, y\rangle = a^* \langle x, y\rangle,\qquad \langle x,y\rangle^* = \langle y,x\rangle \] for all $x,y\in X$ and $a\in \mathcal{A}$;
  and such that $X$ is complete with respect to the induced norm $\|x\|:= \| \langle x, x \rangle \|_A^{1/2}$.
  Given $f,h\in C_c(E^1)$ and $g\in C_0(E^0)$ define:
  • $\bullet$ a right action of $C_c(E^1)$ by $C_0(E^0)$ via $f\mapsto f\cdot g$ where \[ (f\cdot g)(e) = f(e) g(s(e)) \qquad e\in E^1; \]
  • $\bullet$ a map $\phi\colon C_0(E^0)\to \mathcal{L}(C_c(E^1))$ by \[ [\phi(g)f](e) = g(r(e)) f(e) \qquad e\in E^1; \]
  • $\bullet$ a map $\langle\cdot,\cdot\rangle\colon C_c(E^1)\times C_c(E^1)\to C_0(E^0)$ by \[ [\langle f,h\rangle](v) = \sum_{e\in s^{-1}(v)} \overline{f(e)} h(e). \]
  Verify that the "appropriate" quotient and completion of $C_c(E^1)$ is a $C_0(E^0)$-correspondence.

In the second lecture, we will define another classical structure from which we can build a C*-algebra: a graph correspondence. From a graph correspondence, we will construct a Cuntz-Pimsner algebra and discuss its relationship to naturally-associated graph and Cuntz-Krieger algebras. To end, we will create a bird’s-eye-view diagram of the three C*-algebra constructions discussed to illustrate their overlap and examples of when they do not.