Seminar in operator theory and operator algebras (MATH 9310)
Fall 2017

The seminar, organized by David Sherman, meets Tuesdays 4-5 in Kerchof 326.

September 5

Kristin Courtney, UVa

Nilpotent approximations of universal operators and some conjectures

Because of its elegance and utility, von Neumann's inequality has become canon in operator theory, and its extensions to various contexts are still the subject of a wide range of research. The inequality says that, given any polynomial p in one variable, the maximal norm of the operator p(T), as T ranges over all contractive Hilbert space operators, can be determined by considering only contractive operators on a one dimensional Hilbert space, i.e. elements of the unit disk in the complex plane.

Using universal C*-algebras, one can readily show a von Neumann-type inequality for non-commutative *-polynomials, which says that, given a non-commutative *-polynomial q, the maximal norm of the operator q(T), as T ranges over contractive Hilbert space operators, can be determined by considering only contractive operators on finite-dimensional Hilbert spaces, i.e. matrices of norm at most 1. Our first goal in this talk is to show why it actually suffices to consider only nilpotent matrices of norm at most 1.

Moving to polynomials in two variables, von Neumann's inequality notably extends when the argument ranges over pairs of commuting contractive Hilbert space operators. Can we again look to matrices for a bound for the norm of any noncommutative *-polynomial in two variables whose inputs are two doubly commuting contractive operators? Does it suffice to consider only nilpotent matrices? Surely these questions are not too difficult, are they?

September 12

Brian Lins, Hampden-Sydney College

The Illumination Conjecture and fixed points of nonexpansive maps

The Illumination Conjecture is a famous unsolved conjecture in combinatorial geometry. It predicts that the surface of any convex body in Rn can be completely illuminated by 2n floodlights. Surprisingly, it has not been proven, even in 3-dimensions! This talk will focus on a new connection between this famous conjecture and the fixed points of nonexpansive maps in finite dimensional normed spaces.

September 19

no meeting

September 26

no meeting

October 3

READING DAY (no meeting)

October 10

Ben Hayes, UVa

Algebraic actions of sofic groups and their entropy I: general background

I will discuss the notion of an algebraic action, which is an action of a discrete group by automorphisms on a compact group. I will be particularly focused on the discussed of the entropy of such an action for the case when the acting group is sofic (soficity will be defined in the talk).

October 17

Ben Hayes, UVa

Algebraic actions of sofic groups and their entropy II: Fuglede-Kadison determinants and entropy

Continuing on the previous lecture, I will discuss entropy for a large class of algebraic actions and how this relates to von Neumann algebra invariants.

October 24

Geoff Price, US Naval Academy

On the classification of binary shifts

A spin system is a sequence of Hermitian unitary operators u_1, u_2,... that pairwise commute or anticommute. Under a mild condition the von Neumann algebra generated by a spin system is the hyperfinite factor R. Our focus is on spin systems with translation-invariant commutation relations, allowing for the construction of unital *-homomorphisms on R. These binary shifts are uniquely determined by the conditions alpha(u_i)=u_{i+1}. We discuss the classification problem of binary shifts, and show how this problem is related to results on the ranks of sequences of Toeplitz matrices over GF(2).

The Virginia Operator Theory and Complex Analysis Meeting (VOTCAM) will be held at UVa on Saturday October 28. Website

October 31

maybe too spooky for a meeting

November 7

no seminar

November 14

no seminar

November 21

Last day before Thanksgiving break (no meeting)

November 28

Sarah Browne, Penn State

E-theory spectrum

E-theory is an invariant of C*-algebras and in particular is a collection of abelian groups defined in terms of homotopy classes of certain morphisms of C*-algebras. This makes it a natural object to define in terms of stable homotopy groups. In my talk I will detail the notion of E-theory and the framework we require, namely a spectrum of quasi-topological spaces, to represent the E-theory groups as a stable homotopy theory. Then I will highlight how we encode E-theory properties into this construction.

December 5

You can reminisce about previous semesters at the links below:
Fall 2016 Spring 2016 Fall 2015 Spring 2015 Fall 2014 Spring 2014 Fall 2013 Spring 2013 Fall 2012 Spring 2012 Fall 2011 Spring 2011 Fall 2010 Spring 2010 Fall 2009 Spring 2009 Fall 2008 Spring 2008 Fall 2007 Spring 2007 Fall 2006 Spring 2006 Fall 2005 Spring 2005 Fall 2004 Spring 2004 Fall 2003 Spring 2003 Fall 2002