A completely positive map between C*-algebras is a linear map that preserves the ordering of self-adjoint elements on the original C*-algebra and its matrix amplifications. In this talk, I will introduce and motivate the fundamental theory of completely positive maps on C*-algebras. The main goal is to show that all unital completely positive maps on C*-algebras arise as the compression of a $*$-homomorphism.

Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ be C*-algebras throughout.

• 1. Prove the following:
  • (a) If $\phi:\mathcal{A}\to\mathcal{B}$ is a $*$-homomorphism, then $\phi$ is completely positive.
  • (b) If $\phi:\mathcal{A}\to\mathcal{B}$ and $\psi:\mathcal{B}\to\mathcal{C}$ are completely positive, then $\psi\circ\phi$ is completely positive.
• 2. Suppose $\phi:\mathcal{A}\to\mathcal{B}$ is a positive (linear) map. Prove that $\phi$ is $*$-preserving (or self-adjoint), i.e., $\phi(a^*)=\phi(a)^*$ for all $a\in \mathcal{A}.$
• 3. Describe in words how (the proof of) Stinespring’s Dilation Theorem generalizes the GNS construction. [Hint: When $\phi$ is a state, what is $\mathcal{H}$? $\mathcal{A}\odot\mathcal{H}$? $V$?]
• 4. Let $\mathcal{A}$ be a unital C*-algebra and $\phi:\mathcal{A}\to\mathcal{B}$ be a completely positive map. Prove that $\phi(a)^*\phi(a)\le \|\phi\|\phi(a^*a)$ for all $a\in \mathcal{A}$. [Remark: this result also holds in the non-unital case.]
• 5. Let $\mathcal{A}$ be a unital C*-algebra and let $a$ and $b$ be elements in $\mathcal{A}$. Show that \[ \left(\begin{array}{cc} 1 & a\\ a^* & b \end{array}\right)\] is positive in $M_2(\mathcal{A})$ if and only if $a^*a\le b$ in $\mathcal{A}.$

Completely positive maps are fundamental to the study of C*-algebras (and W*-algebras), particularly the study of tensor products of C*-algebras. It turns out that completely positive maps are the correct morphisms for the category of operator systems, which can be realized as unital, self-adjoint subspaces of C*-algebras. In this talk, we will motivate and discuss various non-C*-substructures of C*-algebras.

This talk will introduce Hilbert bimodules as the analogue of representations of groups. We will discuss weak containment as a natural type of approximation and the associated Fell topology.

• 1. Show that weak containment of bimodules is transitive. That is, show that if $_M\mathcal{H}_N \prec {}_M\mathcal{K}_N$ and $_M\mathcal{K}_N \prec {}_M\mathcal{L}_N$ then $_M\mathcal{H}_N \prec {}_M\mathcal{L}_N$.
• 2. Suppose $(M,\tau)$ is a tracial von Neumann algebra. Show that $L^2(M)\subset {}_M\mathcal{H}_M$ if and only if $\mathcal{H}$ contains a tracial, central vector. If $M$ is a factor show that $L^2(M)\subset {}_M\mathcal{H}_M$ if and only if $\mathcal{H}$ contains a central vector.
• 3. Show that $\mathbb{Z}$ is amenable.
• 4. Show that finite groups have property (T).

This talk will define amenability and property (T) in terms of representations and bimodules. The talk will mostly consist of examples, intuition, and fun facts about amenability and property (T).

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 algebra over $A$ along with the Grothendieck construction of an abelian group from an abelian semigroup. A number of concrete examples will be discussed.

Note: reviewing Chapter 8 of the GOALS C*-algebra notes should be helpful.

• 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 $u$ and $v$ are unitaries, then $||u-v||\le 2$.
  • (c) Prove that if $p$ and $q$ are projections, then $||p-q||\le 1$.
  • (d) Give examples of projections and unitaries such that $||u-v||=2$ and $||p-q||=1$.
• 2. Suppose $A$ is a finite dimensional C*-algebra. Compute it's $K$-theory. Hint: This requires a general theorem about direct sums (which you should try to prove) and a theorem about the structure of a finite dimensional C*-algebra).
• 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.

After introducing $K_1(A)$, I will discuss how $K_0$ and $K_1$ are related via a six-term exact sequence. A variants of this six-term exact sequence will be applied to compute the K-theory groups of the crossed product $C^*$-algebra associated with the odometer action on the Cantor set.

In this lecture, we will explore the concept of index for subfactors. We will cover the basic definition of the index and use the local index formula to prove some basic results about the relative commutant. We will briefly talk about the Jones Index theorem and how some of the obstructions for the index arise.

• 1. Let $N\subset M$ be an inclusion of $\mathrm{II}_1$ factors. Show that $\dim(N'\cap M) \leq [M\colon N]$.
• 2. Let $N\subset B(\mathcal{H})$ be a $\mathrm{II}_1$ factor. For $d\in \mathbb{N}$, embed $N\hookrightarrow M_d(N)$ by \[ x\mapsto \left(\begin{array}{ccc} x & & 0 \\ & \ddots& \\ 0 & & x \end{array}\right) \qquad x\in N. \] In this exercise, you will 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 $i,j,k,\ell =1,\ldots, d$.
    [Hint: first show that $L^2(M_d(N)) \cong L^2(N)^{\oplus d^2}$.]
  • (b) Show that $N'\cap B(L^2(M_d(N))) = M_{d^2}(N'\cap L^2(N))$.
  • (c) For $X=(x_{i,j})_{i,j=1}^d\in M_d(N)$, show that \[ e_N X = \left(\begin{array}{ccc} \frac{1}{d} \sum_{i=1}^d x_{i,i} & & 0 \\ &\ddots & \\ 0 & & \frac{1}{d}\sum_{i=1}^d x_{i,i} \end{array} \right). \] as vectors in $L^2(M_d(N))$.
  • (d) Viewing $e_N\in M_{d^2}(N'\cap L^2(N))$, 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 $[M_d(N):N]$.
• 3. Let $\Gamma \overset{\alpha}{\curvearrowright} L^\infty(X,\mu)$ be a free ergodic p.m.p action of a countably infinite discrete group on a probability space $(X,\mu)$. Let $\Lambda< \Gamma$ be a finite index subgroup with \[ \Gamma = \Lambda \sqcup \Lambda g_2\sqcup \cdots \sqcup \Lambda g_n. \] for some $g_2,\ldots, g_n\in \Gamma\setminus\Lambda$. Assume $\alpha|_\Lambda$ is ergodic and set \begin{align*} M &:=L^\infty(X,\mu)\rtimes_\alpha \Gamma\\ N &:=L^\infty(X,\mu)\rtimes_{\alpha|_\Lambda}\Lambda. \end{align*} Recall that $L^2(M)=\ell^2(\Gamma)\otimes L^2(X,\mu)$ and $L^2(N)=\ell^2(\Lambda)\otimes L^2(X,\mu)$.
  • (a) For each $i=2,\ldots, n$, show that $\ell^2(\Lambda g_i)\otimes L^2(X,\mu)$ is reducing for $N$.
  • (b) Let $J$ be the canonical conjugation operator on $L^2(M)$: $J\hat{x}=\widehat{x^*}$. Show that \[ J(\delta_g\otimes f) = \delta_{g^{-1}}\otimes \alpha_{g^{-1}}(\bar{f}) \] for $g\in \Gamma$ and $f\in L^\infty(X,\mu)$.
  • (c) For each $i=2,\ldots, n$, show that $J\lambda(g_i^{-1})J e_N J\lambda(g_i)J\in N'$ and that this is the projection onto the subspace $\ell^2(\Lambda g_i)\otimes L^2(X,\mu)$.
  • (d) For each $i=2,\ldots, n$, show that $e_N$ is equivalent to $J\lambda(g_i^{-1})J e_N J\lambda(g_i)J$ in $N'$.
  • (e) Compute $\tau_{N'}(e_N)$ and $[M:N]$.
• 4. Let $\Gamma$ be an i.c.c. group, let $\Lambda<\Gamma$ be a finite index subgroup, and set $M:=L(\Gamma)$ and $N:=L(\Lambda)$.
  • (a) Show that $\Lambda$ is i.c.c.
  • (b) Suppose $\Gamma=\Lambda\sqcup \Lambda g_2\sqcup \cdots \sqcup \Lambda g_n$ for $g_2,\ldots, g_n\in \Lambda\setminus \Lambda$. For each $i=2,\ldots, n$, show that $J\lambda(g_i^{-1})J e_N J\lambda(g_i)J\in N'$ and that this is the projection onto $\ell^2(\Lambda g_i)$.
  • (c) For each $i=2,\ldots, n$, show that $e_N$ is equivalent to $J\lambda(g_i^{-1})J e_N J\lambda(g_i)J$ in $N'$.
  • (d) Compute $\tau_{N'}(e_N)$ and $[M:N]$.

In this lecture, we will look at the standard invariant for subfactors and its role in the classification for small index subfactors. After, we will introduce the principal and dual principal graph of a subfactor and explain how it relates to the standard invariant. The principal graph for the cross product will be computed and we will describe its dual principal graph. Finally, we will discuss what kind of obstructions show up on principal graphs.