## Operator Algebras Seminar Fall 2016

Organizers: Marius Dadarlat, Andrew Toms, and Thomas Sinclair

Thursday, 2:30-3:30pm, MATH 731

• Date: 8/25
• Title:

• Date: 9/1
• Title:

• Date: 9/8
• Rachel Norton, University of Iowa
• Title: Comparing Nevanlinna-Pick Theorems
• Since the original proof of the Nevanlinna-Pick theorem in 1915, there have been a variety of generalizations to operator theory, all but two of which may be recovered by Muhly and Solel's result from 2004. Muhly and Solel think of Nevanlinna-Pick interpolation as an instance of commutant lifting. Constantinescu, Johnson, and Popescu, on the other hand, use the displacement equation to prove results which are fundamentally different from Muhly and Solel's. In this talk, we address the differences and discuss circumstances under which the theorems are equivalent.

• Date: 9/15
• Title:

• Date: 9/17-18, **Wabash Annual Mini-Conference, IUPUI **

• Date: 9/22
• Title:

• Date: 9/29
• Ben Hayes, Vanderbilt University
• Title: Weak equivalence to Bernoulli shifts for some algebraic actions
• Let $G$ be a countable discrete group. Given two pmp actions of $G$ on $(X,\mu)$, $(Y,\nu)$. There is a way to say that the action of G on $(X,\mu)$ is weakly contained in the action of $G$ on $(Y,\nu)$ due to Kechris (this is formulated in a way similar to weak containment of representations and has similarities to finite representability of Banach spaces). We say that two actions are weakly equivalent if each if weakly contained in the other. An algebraic action of $G$ is an action by automorphisms on a compact, metrizable, abelian group $X$, this is a pmp action if we give $X$ the Haar measure. I'll discuss weak equivalence for certain algebraic actions which are related to convolution operators on $G$. I'll show that when such operators are invertible (equivalent when they are invertible in the group von Neumann algebra), then these actions are weakly contained in Bernoulli shifts. No knowledge of weak containment or algebraic actions will be assumed.

• Date: 10/1-2, **ECOAS 2016, Loyola University**

• Date: 10/6
• Title:

• Date: 10/13
• Title:

• Date: 10/20 **REC 108**
• Sujan Pant, University of Iowa
• Title: Structural results for von Neumann algebras arising from poly-hyperbolic groups and Burger-Mozes groups
• Denote by $\mathcal C$ the class of all non-amenable groups that are hyperbolic of non-trivial free products. For every positive integer $n$ denote by $Quot_n(\mathcal C)$ the class of groups that can be realized as $n$-step extensions of groups in $\mathcal C$ [DSP'16]. We show that the von Neumann algebras of these groups enjoy the following structural property: Let $\Gamma\in Quot_n(\mathcal{C})$ and suppose $A_1,A_2,\ldots, A_k \subset L(\Gamma)$ are $arbitrary$ commuting subalgebras with no amenable direct summands that generate together a finite Pimsner-Popa index of $L(\Gamma)$. Then $\Gamma$ is commensurable to a product $\Lambda_1\times\Lambda_2\times \cdots\times \Lambda_k$ with $\Lambda_i\in Quot_{n_i}(\mathcal{C})$ and $n_1+n_2+\cdots+ n_k=n$. Also, up to corners, $A_i\cong L(\Lambda_i)$, for all $i$. In particular, $L(\Gamma)$ is prime if and only if $\Gamma$ is virtually indecomposible as a product over groups in $Quot(\mathcal{C})$. Our new result strengthens significantly the previous results from [CKP15] by completely removing the exactness and the usage of quasi-cohomological information on $\Gamma$.
The same techniques also show that the von Neumann algebras associated with Burger-Mozes groups gives rise to prime von Neumann algebras. This is the first occurrence of prime factors arising from simple groups. This is based on a joint work with Rolando de Santiago.
[CKP15] I. Chifan, Y. Kida, S. Pant, Primeness results for von Neumann algebras associated with surface braid groups, Int. Math. Res. Not. Vol. 2016 (2016), no. 16, 4807--4848.
[DSP'16] R. de Santiago, S. Pant, Structural results for von Neumann algebras arising from poly-hyperbolic groups and Burger-Mozes groups, Preprints.

• Date: 10/27
• Florin Boca, University of Illinois at Urbana-Champaign
• Title: Distribution of Eigenvalues in Large Sieve Matrices
• The large sieve matrix provides an estimate for the largest eigenvalue of a certain N by N matrix A*A, where A is a Vandermonde type matrix defined by roots of unity of order at most Q. This talk will discuss some aspects concerning the behavior of the eigenvalues of these (large sieve) matrices when N ~ cQ^2, with Q-->\infty and c>0 constant. In particular, we will be interested in obtaining asymptotic formulas for their moments, and establish the existence of a limiting distribution as a function of c. This is joint work with Maksym Radziwill.

• Date: 10/29, **Wabash Seminar, Wabash College, Crawfordsville, IN**

• Date: 11/3
• Title:

• Date: 11/10
• Laszlo Lempert, Purdue University
• Title: Noncommutative potential theory
• Given a Hilbert space $V$ and, say, an open set $\Omega$ in the complex plane, a hermitian metric on the trivial holomorphic vector bundle $E:\Omega\times V\to \Omega$ is the same thing as a function $P = P^*\to {\rm End} V$ whose values are positive, invertible operators. We propose to view such metrics or operator functions as noncommutative analogs of functions defined on the base $\Omega$, and curvature as the notion corresponding to the Laplace operator. We will discuss noncommutative generalizations of basic results of ordinary potential theory: mean value properties, maximum principle, Harnack inequality, and the solvability of Dirichlet problems.

• Date: 11/17
• Stephen Hardy, Hampden-Sydney College
• Title: Pseudomatricial C*-algebras
• Historically, C*-algebras which are built from finite-dimensional C*-algebras in nice ways have been tractable. For instance, the algebra of compact operators (norm limits of finite rank operators), and the AF C*-algebras (inductive limits of finite-dimensional C*-algebras) are well understood. Instead of norm limits or inductive limits, we study the logical limits of matrix algebras. This idea was first used by Ax to study the logical limits of finite fields. The analogous metric objects were introduced by Goldbring and Lopes, using the continuous logic due to Ben Yaacov, Berenstein, Henson, and Usvyatsov. These logical limits of matrix algebras are called the pseudomatricial C*-algebras. We will explore the finiteness properties and K-theory of pseudomatricial C*-algebras.

• Date: 11/24
• Thanksgiving Break

• Date: 12/1
• Title:

• Date: 12/8
• Title: