Seminar in operator theory and operator algebras (MATH 9310)
Fall 2014


The seminar is organized by David Sherman. We meet Tuesdays 4-5 in Kerchof 326.


September 9

Chris Ramsey, UVa

Automorphisms of the free product of disk algebras

In complex topology there is lovely structure theorem due to Rudin, Ligocka and Tsyganov saying that an automorphism of a product of spaces is just a product of automorphisms in each space and a permutation of the spaces in the product. Such an automorphism of the polydisk naturally produces an isometric automorphism of the free product of disk algebras. We will prove that there are no other isometric automorphisms.

September 16

Chris Ramsey, UVa

Operator algebras and multivariable dynamics

To a dynamical system comprised of a compact Hausdorf space and n proper continuous maps one can associate a universal operator algebra. We will show that this universal operator algebra completely characterizes the dynamical system up to a natural form of conjugacy.

September 23

Kevin Beanland, Washington & Lee University

Stabilized Krivine p's

In the early 1970s Jean-Louis Krivine showed that for each basic sequence (xn) in a Banach space there is a p in the interval [1,∞] so that lp is block finitely represented in (xn). It has since been conjectured by several authors that any `stabilized' Krivine set of a Banach space must be an interval. In this talk we will discuss the history of the problem and explain how to construct a wide ranging collection of counterexamples. This is joint work with Dan Freeman and Pavlos Motakis.

Dietmar Bisch (Vanderbilt) will give the department colloquium on September 25.

September 30

David Sherman, UVa

Majorization and the Schur-Horn theorem

Using my lawn sprinkler, I will explain what it means for one real n-tuple to majorize another. The Schur-Horn theorem says that this is exactly the same as the existence of a self-adjoint matrix with the first tuple as eigenvalue list and the second tuple as diagonal. I'll discuss the theorem and the ongoing quest for generalizations and variations (e.g., Gary Weiss's lecture next week).

This talk will be accessible to nonspecialists.

October 7

Gary Weiss, University of Cincinnati

Diagonality and idempotents, Jasper's frame theory problem, and Schur-Horn theorems

We explore various relations of operators to their diagonals which study we coin diagonality and explain. This will include how to use 0-diagonality (operators with zero diagonal in some basis) to answer a frame theory question equivalent to an idempotent question of Jasper's, for which I will give some background.

Then we will explore recent and past work on Schur Horn theorems on possible diagonals a positive compact operator can have.

I hope to give an overview of some core steps in the development of both. This work was inspired by work of Gohberg, Markus, Arveson, Kadison, Jasper, Fan, Fong, Herrero, and others, which will be explained.

Slides from the talk

October 14

Reading day

October 21

Bill Ross, University of Richmond

Pre-orders, partial orders, and partial isometric matrices

The first part of this talk will be a survey on the topic of partial isometric matrices, dating back to gems of P. Halmos and I. Erdelyi. By a complex analysis gem of S. Takenaka, we will also explore their representation as compressed shifts on spaces of rational functions. The second part of the talk will focus on certain pre-orders and partial orders one can place on the partial isometric matrices. All of this relates to a problem first explored by Bruce Crofoot on classifying the multipliers from one model space to another. Though, for the sake of simplicity, we will focus on the partial isometric matrices, there are versions of these results for partial isometries on Hilbert spaces. They will be alluded to at various times during the talk. This is joint work with Rob Martin, Stephan Garcia, Yi Guo, and Zezhong Chen.

October 28

Craig Kleski, Miami University (Ohio)

Korovkin-type properties for completely positive maps

Let S be an operator system in B(H) and let A be its generated C*-algebra. We give a new characterization of Arveson's unique extension property for unital completely positive maps in terms of rigidity with respect to sequences ucp maps. We also show that if every irreducible representation of a Type I C*-algebra A generated by an operator system S is a boundary representation, then every ucp map on A with codomain A" that fixes S also fixes A. This is related to Arveson's "hyperrigidity conjecture" on a noncommutative version of Saskin's classical result on Choquet boundaries and Korovkin sets. We will discuss the basic ideas and definitions of operator systems, completely positive maps, and noncommutative Choquet theory.

November 4

Trevor Richards, Washington & Lee University

The conformal equivalence of polynomials and finite Blaschke products: A tale of three proofs

For any complex polynomial p of one complex variable, if G is a tract of the polynomial (that is, a component of the set {z:|p(z)| less than r} for some positive r) then it is easy to show that if h is the Riemann map for G, then the composition of p with h is a finite Blaschke product on the unit disk (thus we would say that p is conformally equivalent to a finite Blaschke product on G). I will discuss three different proofs (two analytic and one geometric) going the other direction, showing that every finite Blaschke product is in fact conformally equivalent to some polynomial.

Here are the slides.

VOTCAM will be held at Washington & Lee on November 8.

November 11

Stephan Garcia, Pomona College

Complex symmetric operators: a quick overview

Complex symmetric operators are a surprisingly large class of (typically) non-normal operators that arise frequently at the intersection of complex analysis and operator theory. We highlight a number of examples and several key results. This talk will be accessible to graduate students.

November 18

Victor Kaftal, University of Cincinnati

Linear combinations of projections in operator algebras

Here is the abstract.

Here are the slides.

November 25

Last day before Thanksgiving break

December 2

Stephen Hardy, UVa

Pseudocompact C*-algebras

Certain limits of finite-dimensional C*-algebras (i.e. direct sums of matrix algebras), such as AF and UHF C*-algebras and the compact operators, are particularly well understood. We are interested in the class of logical limits of finite-dimensional C*-algebras, which are called pseudocompact C*-algebras. We will discuss results due to Henson and Moore in the Banach space case, some finiteness properties pseudocompact C*-algebras have, and conclude with some questions.





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