Mathematics Department

# Operator Algebras and Dynamics Seminars

## Spring 2017

All talks are from 3:45-4:45 p.m. in the Seminar room, unless otherwise specified.

• Apr
24
• Finite Dimensional Norm Attaining Elements and Finitely Represented C*-algebras
Kristin Courtney
University of Virginia
Time: 03:45 PM

#### View Abstract

One characterization of residual finite dimensionality for a C*-algebra is that the norm of any element is the supremum of the norms of all of its images under finite dimensional representations of the algebra; for some elements this supremum will actually be achieved at some finite dimensional representation. But when does this hold for all elements of the C*-algebra? Moreover, how can the norms of images of an element under finite-dimensional representations grow as the dimension of the representations grow? In this talk, I will present joint work with Tatiana Shulman which utilizes mapping telescopes of AF C*-algebras to answer these questions. In particular, we prove that every element in the C*-algebra achieves its norm under some finite dimensional representation if and only if the C*-algebra is finitely represented, i.e. has only finite dimensional irreducible representations. Furthermore, we prove that these conditions are equivalent to the absence of a simple AF subquotient.
• Apr
17
• Hereditary properties for C*-inclusions
Time: 03:45 PM

#### View Abstract

A property P for C*-inclusions is said to be "hereditary from above" if $A \subseteq B$ has P whenever $A \subseteq C$ has P and $A \subseteq B \subseteq C$. Likewise, P is "hereditary from below" if $B \subseteq C$ has P whenever $A \subseteq C$ has P and $A \subseteq B \subseteq C$. In this talk, we show by example that regularity is not hereditary from above, and that the pure extension property is not hereditary from below. The former example involves ergodic theory and Kazhdan's property (T), while the latter example involves a 1983 result of Baker & Powers about restricting pure product states on the CAR algebra to the GICAR (= gauge-invariant CAR) algebra.
• Apr
05
• Subgroups of Richard Thompson's group F
Justin Moore
Cornell University
Time: 03:45 PM

#### View Abstract

What is the structure of the finitely generated subgroups of Thompson’s group $F$, equipped with the embeddability relation? This is conjectured to be a well quasi-order. We prove that it contains a chain of length $\epsilon_0 + 1$. This is joint work with Matt Brin and Collin Bleak.
• Mar
27
• Two new theorems about similar matrices
Prof. David Sherman
University of Virginia
Time: 03:45 PM

#### View Abstract

It is well-known that if A and B are Hermitian matrices, AB and BA are similar. Is this still true if A and B are merely normal? A matrix V is called a partial isometry if V*V is an orthogonal projection. Which matrices are similar to partial isometries? I was surprised that these questions were open and even more surprised to do joint work that solves them. I'll explain elements of the solutions, give examples establishing sharpness, and discuss some related ideas. The talk will be accessible to anyone who understands the questions above and the sentence, "Jordan canonical form is a complete similarity invariant for complex matrices."
• Mar
06
• Thompson’s Group F: Representation Theory and Dynamics on the Chabauty Space
Kostya Medynets
USNA
Time: 03:45 PM

#### View Abstract

In the talk, we will discuss the structure of characters for Thompson’s group F and its commutator subgroup D(F). We will then discuss the relation between the structure of characters of D(F) (rather the absence thereof) and the dynamics on its Chabauty space, the space of subgroups of D(F). We will also classify conjugation-invariant minimal subsets of the Chabauty space of D(F).
• Feb
27
• Trivial Centralizers of High Complexity Toeplitz Systems
MIDN 1/C James Talisse
USNA
Time: 03:45 PM

#### View Abstract

Given a dynamical system $(X,G)$, the centralizer consists of all homeomorphisms of $X$ which commute with the action of the group $G$. In this talk we present a construction of Toeplitz systems over $\mathbb{Z}^d$ which have a trivial centralizer. Historically, trivial centralizers arise from zero entropy systems. However this construction can lead to examples of systems with positive entropy and trivial centralizers.
• Feb
13
• Obstruction to Lifting Cocycles on Groupoids and Associated C*-Algebras
Marius Ionescu
USNA
Time: 03:45 PM
• Feb
06
• The Normal Subgroup Theorem for Lattices in Products (Property (T))
Darren Creutz
USNA
Time: 03:45 PM

#### View Abstract

Margulis' Normal Subgroup Theorem states that if Gamma is an irreducible lattice in a higher-rank semisimple Lie group with trivial center then every nontrivial normal subgroup of Gamma has finite index. Moving away from Lie groups, Bader and Shalom proved that the same result holds for lattices in products of arbitrary simple nondiscrete locally compact groups. I will present joint work with Y. Shalom which gives a new proof of this result and explain how it leads into my work on a conjecture of Margulis and Zimmer about the nature of commensurated subgroups of lattices. The proof is in two distinct halves (as was Margulis'): we prove Gamma / N is finite by showing it is both amenable and has Kazhdan's property (T). The first discussed the amenability proof; this talk will present the (T) half. In particular, this talk will be a standalone talk and will not assume knowledge of what was talked about in part one.
• Jan
23
• The Normal Subgroup Theorem for Lattices in Products (Part One -- Amenability)
Darren Creutz
USNA
Time: 03:45 PM

#### View Abstract

Margulis' Normal Subgroup Theorem states that if Gamma is an irreducible lattice in a higher-rank semisimple Lie group with trivial center then every nontrivial normal subgroup of Gamma has finite index. Moving away from Lie groups, Bader and Shalom proved that the same result holds for lattices in products of arbitrary simple nondiscrete locally compact groups. I will present joint work with Y. Shalom which gives a new proof of this result and explain how it leads to my work with J. Peterson that every ergodic action of an irreducible lattice in a product of higher-rank semisimple groups on a nonatomic probability space is essentially free. The proof is in two distinct halves (as was Margulis'): we prove Gamma / N is finite by showing it is both amenable and has Kazhdan's property (T). Part one will discuss the amenability proof; part two (to be scheduled) will discuss (T). Note: part two will not rely on part one; each talk will be stand-alone.