Uncountably Many Non-isomorphic Ultrapowers of II₁ factors

Remi Boutonnet

In this talk I will show that there exist infinitely many (even uncountably many) II₁ factors with non-isomorphic ultrapowers. More precisely, I will construct a concrete family of such factors, whose ultrapowers can be distinguished by computing a new invariant for von Neumann algebras. This invariant relies on the notion of central sequence. This is based on a joint work with Ionut Chifan and Adrian Ioana.

Independence Tuples and Deninger's Problem

Ben Hayes

In 2009, Deninger asked the following question: If $G$ is a countable, discrete group and A is an n $xn$-matrix over the integral group ring and $A$ is invertible in $M_{n}(l^{1}(G))$ is the Fuglede-Kadison determinant of $A$ bigger than $1$? Using our previously established connection between Fuglede-Kadison determinants and sofic entropy, we show that Deninger's question is true if $G$ is sofic and we can in fact allow $A$ to be invertible in $M_{n}(L(G))$ where $L(G)$ is the group von Neumann algebra. Crucial to our proof is a modification of the Independence Tuples for an action of a sofc group on a compact metrizable space defined by Kerr-Li. Our modification takes into account the Koopman representation of the induced action of G on the Loeb measure space. We also use our techniques to show that if G is a sofic group and A is an n xn-matrix over the integral group ring and A is invertible in $M_{n}(L(G))$ then the action G has on the Pontryagin dual of $(Z(G)^{n}/Z(G)^{n}A)$ has completely positive entropy. This gives us more examples of the ``Bernoulli-like" behavior such actions have. No knowledge about sofic groups or sofic entropy will be assumed.

Unique Prime Factorization for Von Neumann Algebras of Equivalence Relations

Daniel Hoff

A tracial von Neumann algebra $M$ is called prime if it cannot be decomposed as the tensor product of two nontrivial (not type ${\rm I}$) subalgebras. Naturally, if $M$ is not prime, one asks if $M$ can be uniquely factored as a tensor product of prime subalgebras. The first result in this direction is due to Ozawa and Popa in 2003, who gave a large class of groups $\mathcal{C}$ such that for any $\Gamma_1, \dots, \Gamma_n \in \mathcal{C}$, the associated von Neumann algebra $L(\Gamma_1) \,\overline{\otimes}\, \cdots \,\overline{\otimes}\, L(\Gamma_n)$ is uniquely factored. This talk will focus on von Neumann algebras arising from a class of measured equivalence relations, and how the techniques of Ozawa and Popa can be adapted to this setting.

Local Spectral Gap in Simple Lie Groups

Adrian Ioana

I will present a "local" spectral gap theorem for translation actions of dense subgroups generated by algebraic matrices on arbitrary simple Lie groups. This extends to the non-compact setting works of Bourgain-Gamburd and Benoist-de Saxce. I will also present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, and spectra of averaging operators on compact groups. This is joint work with Remi Boutonnet and Alireza Salehi-Golsefidy.

Tower Decompositions for Actions of Amenable Groups

David Kerr

Recently Downarowicz, Huczek, and Zhang proved that every amenable discrete group can be tiled by translates of finitely many approximately invariant finite sets. Using this result I will show that for every free probability-measure-preserving action of an amenable discrete group there are decompositions of the space into finitely many Rokhlin towers with approximately invariant shapes, strengthening a theorem of Ornstein and Weiss. I will then discuss applications to topological dynamics and the study of C*-crossed products.

Skein Theory for Subfactors

Zhengwei Liu

We provide two different skein theories to construct subfactors with small and large indices respectively, and we construct a new family of subfactors whose indices approach infinity. The idea of the "universal skein theory" is to simplify the construction of subfactors by the knowledge of the Temperley-Lieb-Jones algebra. When the index is small, we can construct subfactors by solving some simple equations. When the index is large, this method in no longer efficient. Instead, we suggest a different type of skein theory motivated by the Yang-Baxter equation and construct a new family of subfactors.

Determinants in K-theory and Operator Algebras

Joeseph Migler

A determinant in algebraic K-theory is associated to any two Fredholm operators that commute modulo the trace ideal. One can also calculate a homological invariant known as joint torsion. In this talk I will discuss recent work on these invariants and some applications.

Approximate equivalence of group actions

Sorin Popa

We consider several weaker versions of the notion of conjugacy and orbit equivalence of probability measure preserving (pmp) actions of groups, involving equivalence of the ultrapower actions and asymptotic intertwining conditions. We compare them with the other existing equivalence relations between pmp group actions, and study the usual type of rigidity questions around these new concepts: superrigidity, calculation of invariants, etc. Based on joint work with Andreas Aaserud.

Zeta Functions and Local Index Theory

Rudy Rodsphon

Since the works of Connes and Moscovici in 95, it is well known that index theorems may be recovered using residues of zeta functions. The advantage of this approach is to be applicable to more delicate situations than the ones from 'classical' index theorems. In this talk, we shall first give an overview of these methods, then discuss some recent results for transversally elliptic operators on foliations (joint with D. Perrot).

Approximation of Operator Functions

Anna Skripka

The talk will summarize recent results on existence of good approximations of operator functions that are analogous to Taylor approximations of scalar functions. The norm in which the error of an approximation is measured is problem specific, and types of approximating expressions that give small errors (when they exist) depend on types of the selected scalar functions and operators.

Generalized Q-gaussian von Neumann Algebras with Coefficients and Their Structural Properties

Bogdan Udrea

For every sequence of symmetric independent copies $(\pi_j,B,A,D)$, every subset $1 \in S = S^* \subset A$ and every separable Hilbert space $H$ we define the associated generalized $q$-gaussian von Neumann algebra $\Gamma_q(B,S \otimes H)$ with coefficients in $B$. We then prove that under suitable technical assumptions the von Neumann algebra $M=\Gamma_q(B,S \otimes H)$ is strongly solid over $B$, i.e. for every von Neumann subalgebra $\mathcal A \subset M$ which is amenable over $B$, either a corner of $\mathcal A$ embedds into $B$ inside $M$ in the sense of Popa, or the von Neumann algebra generated by the normalizer of $\mathcal A$ in $M$ is amenable relative to $B$ inside $M$. Time permitting, we will also talk about other structural properties of these algebras, such as solidity, embedability into $R^{\omega}$ and absence of non-trivial central sequences. This is joint work with Marius Junge (University of Illinois at Urbana-Champaign).

QDQ vs. UCT

Wilhelm Winter

I will explain a recent joint result with Aaron Tikuisis and Stuart White, saying that faithful traces on separable nuclear C*-algebras which satisfy the universal coefficient theorem are quasidiagonal. This confirms Rosenberg's conjecture that discrete amenable groups have quasidiagonal C*-algebras. It also resolves the Blackadar-Kirchberg problem in the simple UCT case. Moreover, there are several consequences for Elliott's classification programme; in particular, the classification of separable, simple, unital, nuclear, Z-stable C*-algebras with at most one trace and satisfying the UCT is now complete; the invariant in this case is ordered K-theory.