Given a countable discrete group acting on a probability space, one can define the crossed product von Neumann algebra. Also called the group measure space construction, this von Neumann algebra is generated by the abelian von Neumann algebra given by the measure space and the group von Neumann algebra of the countable discrete group. The relation between these two von Neumann algebras inside the crossed product is given by the commutation relation induced by the action of the group. We show that if the action is free and ergodic, then the crossed product von Neumann algebra is a factor.

• 1. Suppose $\Gamma \overset{\alpha}{\curvearrowright} L^\infty(X,\mu)$ for a countable discrete group $\Gamma$ and a probability space $(X,\mu)$.
  • (a) For $f\in L^\infty(X,\mu)$, show that $\pi_\alpha(f)$ is a bounded operator on $\ell^2(\Gamma)\otimes L^2(X,\mu)$ with $\|\pi_\alpha(f)\|=\|f\|_\infty$.
  • (b) Show that $\pi_\alpha \colon L^\infty(X,\mu)\to B(\ell^2(\Gamma)\otimes L^2(X,\mu))$ is a unital $*$-homomorphism.
  • (c) Show that $\lambda(g)\pi_\alpha(f)\lambda(g^{-1}) = \pi_\alpha( \alpha_g(f))$ for all $g\in \Gamma$ and $f\in L^\infty(X,\mu)$.
• 2. Suppose $\Gamma \overset{\alpha}{\curvearrowright} L^\infty(X,\mu)$ for a countable discrete group $\Gamma$ and a probability space $(X,\mu)$. For $f\in L^\infty(X,\mu)$, define $\phi_\alpha(f)\in B(\ell^2(\Gamma)\otimes L^2(X,\mu))$ by \[ \phi_\alpha(f)\left(\sum_{g\in \Gamma} \delta_g\otimes f_g\right) = \sum_{g\in \Gamma} \delta_g\otimes f_g f\qquad f_g\in L^2(X,\mu), \] and define $\rho(g)$ for $g\in \Gamma$ by \[ \rho(g)\left(\sum_{h\in \Gamma} \delta_h\otimes f_h\right) = \sum_{h\in \Gamma} \delta_{hg^{-1}}\otimes \alpha_g(f_h) \qquad f_h\in L^2(X,\mu). \] Show that $\phi_\alpha(L^\infty(X,\mu))\cup \rho(\Gamma)\subset (L^\infty(X,\mu)\rtimes_\alpha \Gamma)'$.
• 3. Suppose $\Gamma \overset{\alpha}{\curvearrowright} L^\infty(X,\mu)$ for a countable discrete group $\Gamma$ and a probability space $(X,\mu)$.
  • (a) Show that $\delta_e\otimes 1$ is a cyclic vector for $L^\infty(X,\mu)\rtimes_\alpha \Gamma$.
  • (b) Show that $\delta_e\otimes 1$ is a separating vector for $L^\infty(X,\mu)\rtimes_\alpha \Gamma$.[Hint: use the previous exercise.]
• 4. In this exercise, you will show that $M_n(\mathbb{C})$ can be realized via a crossed product construction. Consider $\Gamma:=\mathbb{Z}_n$, the cyclic group of order $n$, and also set $X:=\mathbb{Z}_n$ which we view as simply a space and equip with the uniform probability measure.
  • (a) Show that $\alpha_g(f):=f(\,\cdot\,-g)$ for $g\in \Gamma$ defines an action $\Gamma\overset{\alpha}{\curvearrowright} L^\infty(X,\mu)$.
  • (b) Show that $\Gamma\overset{\alpha}{\curvearrowright} L^\infty(X,\mu)$ is free, ergodic, and probability measure preserving.
  • (c) Show that $1_{\{1\}}, \ldots, 1_{\{n\}}\in L^{\infty}(X,\mu)$ are pairwise orthogonal and equivalent minimal projections.
  • (d) Show that $L^\infty(X,\mu)\rtimes_\alpha \Gamma\cong M_n(\mathbb{C}) $. What is the preimage of $E_{i,j}$ under this isomorphism?
  • (e) Explain why there does not exist a discrete group $\Gamma $ such that $L(\Gamma) \cong M_n(\mathbb{C}) $.

Consider the group measure space construction of a countable discrete group acting on a standard probability space. Turns out that if the action is free, then the von Neumann algebra only depends on the equivalence relation induced by the action on the measure space. Given a standard Borel equivalence relation, one can define the equivalence relation von Neumann algebra. We give this definition and mention Feldman and Moore's result about Cartan inclusions.

In this talk, we will talk about the basic construction of a C*-algebra from a directed graph using generators and relations. We will discuss the Cuntz-Krieger and gauge-invariant uniqueness theorems, and conclude with a discussion on Condition (K) and ideals in graph algebras.

• 1. For edges $e,f$ in $E^1$, when is $s_es_f\in C^*(E)$ nonzero? Prove that $s_es_f$ is always a partial isometry. Compare the range and source projections of $s_es_f, s_e$, and $s_f$.
• 2. Given any two edges $e,f$ in $E$, we have $s_es_e^*s_fs_f^*=\delta_{e,f} s_es_e^*$ in $C^*(E)$. More generally, $s_e^*s_f=\delta_{e,f}p_{s(e)}$. Conclude that $$C^*(E)=\overline{\text{span}}\{s_\lambda s_\mu^*: \lambda = \lambda_1\cdots \lambda_n, \mu=\mu_1\cdots\mu_m \text{ are paths in $E$}\}$$
• 3. Consider the following graph $F$:
  Prove that $C^*(F)\cong M_4(\mathbb{C})$. More generally, prove that if $E$ is a finite graph with no cycles, then $$C^*(E)\cong \bigoplus_{\text{sources $v \in E^0$}}M_{n(v)}(\mathbb{C})$$ where $n(v)$ is the number of paths in $E$ starting at $v$ (including the length-zero path $v$).
  [Hint: Use the previous exercise; you may further assume that $C^*(E)$ is densely spanned by $s_\lambda s_\mu^*$ where $s(\lambda)=s(\mu)$.]
  [Hint: Use the second Cuntz--Krieger relation $p_w=\sum_{e \in r^{-1}(w)}s_es_e^*$.]
• 4. Let $E$ be the following graph: Prove that $C^*(E)$ is isomorphic to the Toeplitz algebra $\mathcal{T}$. [Hint: Consider the isometry $s_e+s_f$.]
• 5. Let $E$ be the following graph: Verify that $E$ satisfies Condition (K). Construct the ideal lattice of $C^*(E)$.

Hilbert C*-modules are an important gadget in C*-algebras with a wide variety of applications. After introducing Hilbert modules, we will discuss how the Cuntz-Pimsner algebra is obtained from a C*-correspondence (Hilbert bi-module), and how this is used to define the C*-algebra of a topological graph.

I will discuss the construction of an abelian group denote $K_0(A)$ from a unital C*-algebra, $A$. The construction involves projections in the matrix algebras over $A$ along with the Grothendieck construction of an abelian group from an abelian semigroup. A number of concrete examples will be discussed.

• 1. Suppose $A$ is a unital $C^*$-algebra.
  • (a) Prove that if $u$ and $v$ are unitaries, then $||u-v||\le 2$.
  • (b) Prove that if $p$ and $q$ are projections, then $||p-q||\le 1$.
  • (c) Give examples of unitaries and projections such that $||u-v||=2$ and $||p-q||=1$.
• 2. Suppose $A$ is a finite dimensional $C^*$-algebra. Compute $K_0(A)$.
  [Hint: This requires a general theorem about the $K_0$-group of direct sums (which you should try to prove) and a theorem about the structure of a finite dimensional $C^*$-algebra (which appears in Chapter 8 of the GOALS $C^*$-algebra notes).]
• 3. Compute the $K$-theory of the following $C^*$-algebras:
  • (a) the compact operators;
  • (b) the CAR algebra;
  • (c) $C(X)$ where $X$ is the Cantor set. A specific realization of the Cantor set is \[ \{ (a_0, a_1, \ldots ) \mid a_i \in \{ 0, 1\} \} \] where the relevant topology is the product topology.
  [Hint: start by writing each algebra as an inductive limit of finite dimensional algebras.]

After introducing $K_1(A)$, I will discuss how $K_0$ and $K_1$ are related via a six-term exact sequence. A number of (other) six-term exact sequences will be introduced and some computations will be discussed. For example, we will compute the K-theory groups of the crossed product $C^*$-algebra associated with the odometer action on the Cantor set.

We motivate the study of subfactors and zoom in on some basic examples (which, alas, arise from finite group theory) and the tools—the index and basic construction—which give us our first handles on subfactors.

• 1. Prove that factor von Neumann algebras have no 2-sided ideals that are also $\sigma$-WOT closed.
• 2. If $f: A \rightarrow B$ is a *-homomorphism of von Neumann algebras, show that its kernel is a $\sigma$-WOT closed 2-sided ideal.
• 3. Embed $N$, a II$_1$ factor, diagonally in $M_d(N)$. We wish to compute $[M_d(N):N]$.
  • (a) Show that $B(L^2(M_d(N))) = M_{d^2}(B(L^2(N)))$, where the entries in the latter space are indexed by pairs of pairs: $((i,j),(k,\ell))$ for $i,j,k,\ell = 1,...,d$. [Hint: first show that $L^2(M_d(N)) \simeq L^2(N)^{\oplus d^2}$ .]
  • (b) Show that $N' \cap B(L^2(M_d(N))) = M_{d^2} (N' \cap B(L^2(N)))$.
  • (c) For $X = (x_{i,j})_{i,j=1}^d \in M_d(N)$, show that \[ e_N X = \left( \begin{array}{lll} \frac{1}{d} \sum_{i=1}^d x_{i,i} & & 0 \\ & \ddots & \\ 0 & & \frac{1}{d} \sum_{i=1}^d x_{i,i} \end{array} \right) \] as vectors in $L^2(M_d(N))$.
  • (d) Viewing $e_N \in M_{d^2} (N' \cap B(L^2(N)))$,show that the $((i,j),(k,\ell))$-entry of $e_N$ is $\frac{1}{d} \delta_{i=j} \delta_{k = \ell}$.
  • (e) Compute $\tau_{M_d(N)} (e_N )$ and $[M_d(N) : N]$.
• 4. Show that the basic construction for $N \subset M_d(N)$ is $M_{d^2} (N)$. [Hint: You computed $e_N$ in the previous problem. Make use of the fact that, for $N \subset M$, we have that $\langle M, e_N \rangle$ is generated by the *-algebra which is span$(M \cup M e_N M)$.]
• 5. The quantum integers are defined as \[ [n]_q := \displaystyle \frac{q^n-q^{-n}}{q-q^{-1}} = q^{n-1} + q^{n-3} + \cdots + q^{-n+3} + q^{-n+1}. \]
  • (a) Prove that $[2]_q [n]_q = [n+1]_q + [n-1]_q$.
  • (b) Prove that $[4]_q [3]_q = [6]_q + [4]_q + [2]_q$.
  • (c) Can you write $[5]_q [3]_q$ as a sum of quantum integers?
  • (d) For what values of $q$ does $[n]_q=0$?

We will outline a proof of Jones’ index theorem and delve into the role of the Temperley–Lieb agebra, as  well as some more interesting examples of subfactors.