Crossed products appear frequently in the literature and so it is essential for beginning researchers in operator algebras to understand their construction and uses.They arise from dynamical systems (most commonly from time evolution systems) and encode the action of a group on a C*-algebra into a new C*-algebra.
In our first talk we discuss the building blocks we will need to construct crossed products of C*-algebras. We begin with the definitions of topological groups and locally compact groups. Then we look at the actions of a group $\Gamma$ on a C*-algebra $A$, which is a group homomorphism $\alpha: \Gamma \to \text{Aut}(A)$. Next, after a brief review of Haar measures, we investigate the Haar integral and convolution products. From this we are able to define covariant representations $(u,\pi,\mathcal{H})$ which we use to assign a norm on the continuous functions of compact support from a group $\Gamma$ to a C*-algebra $A$ denoted $C_c(\Gamma, A)$. Once $C_c(\Gamma,A)$ is completed under this norm we arrive at the crossed product C*-algebra $A \rtimes_{\alpha,\pi}\Gamma$.

• 1. (Review of semi-direct products): Let $H$ and $K$ be groups and suppose that we have a group homomorphism $\phi:K \to \text{Aut}(H)$. Let $G$ be the set of ordered pairs $(h,k)$ with $h\in H$ and $k\in K$. Define multiplication on $G$ by \[ (h_1,k_1)(h_2,k_2) := (h_1\phi_{k_1}(h_2),k_1k_2) \] and inverses by \[ (h,k)^{-1} := (\phi_{k^{-1}}(h^{-1}),k^{-1} ). \] This makes $G$ a group with identity $(1,1)$. Embed $H$ and $K$ into $G$ via the maps \[ h \mapsto (h,1) \quad\quad \text{and} \quad\quad k \mapsto (1,k) \]
  • (a) Using these embeddings show that $K$ acts on $H$ by conjugation.
  • (b) Recall that the normalizer of a group, ring, or algebra $G$ for a subset $S\subseteq G$ is defined as \[ N_G(S) := \{g\in G:gSg^{-1} = S\}. \] Show that $K\leq N_G(H)$. Why does this show that $H$ is normal in $G$?
  • (c) Let $(u,\pi,\mathcal{H})$ be a covariant representation of the dynamical system $(A,\Gamma,\alpha)$ (you may assume that $\Gamma$ is discrete). Show that $u(\Gamma)$ is a subgroup of $N_{(A \rtimes_{\alpha,\pi} \Gamma)}(\pi(A))$
  • (d) Show that $\pi(A)$ is a subalgebra of $N_{(A \rtimes_{\alpha,\pi} \Gamma)}(\pi(A))$
  • (e) Note that $A \rtimes_{\alpha,\pi} \Gamma$ is generated by $u(\Gamma)$ and $\pi(A)$. What might prevent $A \rtimes_{\alpha,\pi} \Gamma$ from being in the normalizer of $\pi(A)$ as a subalgebra?
• 2. Let $(A, \Gamma, \alpha)$ be a C*-dynamical system with $\Gamma$ discrete. Recall that multiplication on $C_c(\Gamma,A)$ is defined by \[ f *_{\alpha}g := \sum_{s,r\in \Gamma}f_r\alpha_r(g_{r^{-1}s})s \] and involution is defined by \[ f^* := \sum_{s\in \Gamma} \alpha_{s^{-1}}(f_s^*)s^{-1}. \]
  • (a) Show that this makes $C_c(\Gamma,A)$ a $*$-algebra.
  • (b) Suppose that $A = \mathbb{C}$. Show that $C_c(\Gamma,\mathbb{C}) = \mathbb{C}[\Gamma]$ as $*$-algebras.
• 3. Recall that if $\mu$ is a Haar measure on a locally compact group $\Gamma$ then there is a continuous homomorphism $\Delta: \Gamma \to \mathbb{R}$ such that \[ \Delta(r) \int_{\Gamma}f(sr)d\mu(s) = \int_{\Gamma} f(s) d\mu(s) \qquad \forall f \in C_c(\Gamma,A) \] If $\Gamma$ is discrete show that $\Delta \equiv 1$.
• 4. Let $(u,\pi,\mathcal{H})$ be a covariant representation of the C*-dynamical system $(A,\Gamma,\alpha)$. Show that the integrated form of $(u,\pi,\mathcal{H})$ is a $L^1$-norm decreasing $*$-representation of $C_c(\Gamma,A)$.

In our second talk we look at the two most common completions of $C_c(\Gamma, A)$. The first being the universal crossed product $A \rtimes_{\alpha,u}\Gamma$ which is obtained from the supremum norm; that is, the supremum of all of the norms obtained from covariant representations. Once we have constructed the universal crossed product we prove the universal property that it enjoys. Next, we construct the regular representation by using a faithful representation and acting on it spatialy by the left regular representation to obtain a covariant representation. Completing $C_c(\Gamma,A)$ under the norm obtained from this representation we have the reduced crossed product $A \rtimes_{\alpha,r}\Gamma$. Lastly, we show that the reduced crossed product does not depend on the choice of faithful representation.

We will learn some of the basic definitions in free probability, including non-commutative probability spaces, non-commutative distributions, and the central notion of free independence. We will cover the combinatorial view of free independence and together use it to prove the free central limit theorem (modulo a few facts we'll take for granted).

• 1. (Finishing proof of Free CLT; doing hands-on moment calculation): Suppose $(a_i)_{i \in \mathbb{N}}$ is a family of self-adjoint, freely independent, identically distributed nc random variables with $\phi(a_i) = 0$ and $\phi(a_i^2) = \sigma^2$. Compute the following:
  • (a) $\phi(a_1 a_2 a_3)$
  • (b) $\phi(a_1 a_2 a_1)$
  • (c) $\phi(a_1 a_1 a_2 a_2)$
  • (d) $\phi(a_1 a_2 a_1 a_2)$
  • (e) $\phi(a_1 a_2 a_2 a_1)$
  • (f) Generalize the process in (3) and (5) above to arbitrary even-length products with 2 of each index, such as $\phi(a_1 a_2 a_3 a_3 a_2 a_1)$.
    For $\pi \in \mathcal{P}_2(2n)$, what conditions are needed to get $\phi(\pi) = \sigma^{2n}$? What about to get $\phi(\pi) = 0$? Are there any other possible values for $\phi(\pi)$?
• 2. (Applying relationship between moment & cumulant functionals): Show that if $\phi$ is a trace, then the cumulants $\kappa_n$ are invariant under cyclic permutations, i.e. \[ \kappa_n(a_1, a_2, \ldots, a_n) = \kappa_n(a_2, a_3, \ldots, a_n, a_1). \]

After finishing up the proof of the free central limit theorem, we will see how free probability plays a role in a major result related to the free group factor isomorphism problem. Then we'll learn the basic ideas behind the two main notions of free entropy, microstates and non-microstates.

I'll explain how to formulate and prove the Toeplitz index theorem - a formula equating to the number of solutions to certain linear equations with a topological invariant - and why this 'should' be done using K-theory. In the process, I'll introduce K-theory for C*-algebras.

• 1. Let $A$ be a unital $C^*$-algebra, which we assume is (unitally) contained in $\mathcal{B}(H)$ for some Hilbert space $H$. The goal of this exercise is to show that every idempotent in $A$ is connected by a path (through other idempotents!) to a projection in $A$.
  • (a) Let $e\in A$ be an idempotent. Write \[ H=\text{im}(e)\oplus \text{im}(e)^\perp. \] Show that with respect to this decomposition \[ e=\begin{pmatrix} 1 & a \\ 0 & 0 \end{pmatrix}. \] (Put differently, this means $e=p+b$, where $p=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ is the projection onto $\text{im}(e)$, and $b=\begin{pmatrix} 0 & a \\ 0 & 0 \end{pmatrix}$ is a bounded operator that takes $\text{im}(e)^\perp$ into $\text{im}(e)$ and takes $\text{im}(e)$ to zero).
  • (b) Where is the gap in the following argument?

    "Let $e\in A$ be an idempotent, and write $e=\begin{pmatrix} 1 & a \\ 0 & 0 \end{pmatrix}$ as in part (a). For $t\in [0,1]$, define $e_t:=\begin{pmatrix} 1 & ta \\ 0 & 0 \end{pmatrix}$. The path $(e_t)_{t\in [0,1]}$ shows that $e\sim p$."

    [Hint: what's missing is the solution to part (d)...]
  • (c) Let $e$ be an idempotent in a unital $C^*$-algebra. Show that \[ z:=1+(e-e^*)(e^*-e) \] is invertible.
    [Hint: $(e-e^*)(e^*-e)=(e-e^*)(e-e^*)^*$.]
  • (d) Show that if $e=p+b$ is as in part (a), then $p=ee^*z^{-1}$. Conclude that $p$ is in the $C^*$-algebra $A$ (more precisely, it is in the $C^*$-algebra generated by $e$ and the unit of $A$).
    [Hint: $p=ee^*z^{-1}$ is equivalent to $pz=ee^*$. I recommend doing the computations using matrices.]
  • (Bonus) Show that a non-zero idempotent $e$ in a $C^*$-algebra is a projection if and only $\|e\|=1$.
    [Hint: compute the norm of $ee^*$ using a matrix representation as in part (a).]
    Show moreover that $M_2(\mathbb{C})$ contains idempotents with arbitrarily large norm.
• 2. Let $A$ be a (unital) $C^*$-algebra, and consider the following relations on $\mathcal{I}_\infty(A)$.
  • (i) $e\sim_1 f$ if $e$ and $f$ are in the same path component of some $I_n(A)$.
  • (ii) $e\sim_2 f$ if $e$ and $f$ are in the same $\mathcal{I}_n(A)$, and there is $u\in GL_n(A)$ such that $ueu^{-1}=f$.
  • (iii) $I_n(A)\owns e\sim_3 \begin{pmatrix} e & 0 \\ 0 & 0 \end{pmatrix} \in \mathcal{I}_{n+1}(A)$.
  The goal of this problem is to show the equivalence relation on $\mathcal{I}_\infty(A)$ generated by $\sim_1$ and $\sim_3$ is the same as that generated by $\sim_2$ and $\sim_3$.
  • (a) Assume $e,f$ are idempotents in a unital $C^*$-algebra $A$, and $u\in GL(A)$ satisfies $ueu^{-1}=f$. Define \[ u_t:=\begin{pmatrix} \cos(t) & -\sin(t) \\ \sin(t) & \cos(t) \end{pmatrix} \begin{pmatrix} u & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \cos(t) & \sin(t) \\ -\sin(t) & \cos(t) \end{pmatrix}. \] Show that $u_t(e\oplus 0)u_t^{-1}$ is a continuous path connecting $e\oplus 0$ and $f\oplus 0$.
  • (b) Let $e,f$ be idempotents in a unital $C^*$-algebra such that $\|e-f\|<(\|e\|+\|f\|)^{-1}$. Show that $u:=ef+(1-e)(1-f)$ is invertible and satisfies $u^{-1}eu=f$.
    [Hint: show $u$ is invertible by showing that it is close to $1$; having shown this, "$u^{-1}eu=f$" is equivalent to "$eu=uf$".]
  • (c) Convince yourself the previous two problems give the desired conclusion.
    [Hint: with the information given, the harder direction is probably the one starting with a path. Split the path into pieces that are small enough that you can apply the conclusion of part (b).]
  • (Bonus) Explain (preferably "geometrically" in terms of subspaces of the relevant Hilbert space) why the formula $u=ef+(1-e)(1-f)$ is the "right" thing to do in part (b).

The Kadison-Kaplansky conjecture says that if G is a (countable) torsion free discrete group then the reduced group C*-algebra C*_r(G) has no projections other than 0 and 1. I will explain (with very few details!) what the covering index theorem says, and how it is relevant.

We will look at how the notion of property (T) was defined for groups, II$_1$ factors, and tracial von Neumann algebras. We will also mention amenability and its relation with property (T).

• 1. Suppose $G$ is amenable and has property (T). Show that $G$ is finite.
• 2. Show that:
  • (a) finite groups have property (T);
  • (b) the free group on two generators $F_2$ is not amenable;
  • (c) $F_2$ does not have property (T).
• 3. Let $M$ be a II$_1$ factor. Then the standard bimodule $L^2(M)$ has an $M$-central vector. For an $M$-$M$-bimodule $\mathcal{H}$, show that if $\mathcal{H}$ has a nonzero $M$-central vector, then $\mathcal{H}$ contains $L^2(M)$ as a direct summand.
• 4. (Connes and Jones (1985), Proposition 1) Let $(M, \tau)$ be a II$_1$ factor. Suppose $M$ satisfies the following:
  • (i) (property (T)) there exists a finite subset $F_0$ of $M$ and $\varepsilon_0 >0$, such that for any $M$-$M$-bimodule $\mathcal{H}$, if $\mathcal{H}$ has a unit vector $\xi$ with $\max_{x \in F_0} ||x \xi - \xi x|| \le \varepsilon_0$, then $\mathcal{H}$ has a nonzero $M$-central vector;
  • (ii) (the spectral gap property) there exists a finite subset $F_1$ of $\mathcal{U}(M)$ and $c >0$, such that for any $\xi \in L^2(M)$, $c \max_{u \in F_1} ||u\xi - \xi u||_2 \ge ||\xi - \tau(\xi)||_2$ (this is automatic when $M$ has property (T)).
  Let $\mathcal{H}$ be an $M$-$M$-bimodule.
  • (a) Show that $\mathcal{H}$ can be decomposed into $M$-$M$-subbimodules $\mathcal{H} = \mathcal{H}_1 \oplus \mathcal{H}_2$, where $ \mathcal{H}_1$ is a multiple of $L^2(M)$ and $\mathcal{H}_2$ does not contain any nonzero $M$-central vector.
  • (b) For any $\xi \in \mathcal{H}$, write $\xi = \xi_1 + \xi_2$, where $\xi_i \in \mathcal{H}_i$. Show that $||\xi_2|| \le \max_{x \in F_0 \cup F_1} || x \xi -\xi x|| / \varepsilon_0$.
  • (c) Write $\xi_1 = \xi_1' + \xi_1''$, where $\xi_1'$ is $M$-central and $\xi_1''$ is orthogonal to the space of $M$-central vectors. Show $||\xi_1''|| \le c \max_{x \in F_0 \cup F_1} || x \xi -\xi x|| $.
  • (d) Show that for any $\varepsilon > 0$, there exist a finite subset $F$ of $M$ and $\delta > 0$, such that for any $M$-$M$-bimodule $\mathcal{H}$, if $\mathcal{H}$ has a unit vector $\xi$ with $\max_{x \in F} ||x \xi - \xi x|| \le \delta$, then $\mathcal{H}$ has an $M$-central vector $\eta$ with $||\eta - \xi|| \le \varepsilon$.

We will look at some of the first rigidity results on Property (T) for von Neumann algebras in Connes (1980).