Seminar in operator theory and operator algebras
Spring 2008


The seminar is organized by Tom Kriete, and this webpage is maintained by David Sherman. We usually meet Tuesdays from 3:30 to 4:30 in either 326, 205, or 211 Kerchof.



January 17

2:30 PM

(Note special day and time)

Christopher Hammond, Connecticut College

Norm inequalities for composition operators on Hardy and weighted Bergman spaces

It is well known that any analytic self-map of the open unit disk induces a bounded composition operator on the Hardy space and the standard weighted Bergman spaces. For a particular self-map, one might wonder whether there is any meaningful relationship between the norms of the corresponding operators acting on each of these different spaces. After discussing the context of the problem, we will prove an inequality that (at least to a certain degree) answers this question. We will also consider some related questions that are still open, and discuss possible connections between this problem and the issue of cosubnormality of composition operators. (This talk is based on ongoing research with Linda J. Patton.)

January 22

NO SEMINAR - CONCURRENT COLLOQUIUM

January 29

NO SEMINAR - DEPARTMENT MEETING

February 5

Bill Ross, University of Richmond

Truncated Toeplitz operators

In this talk I will make the case for truncated Toeplitz operators - compressions of classical Toeplitz operators on H2 to a model space (invariant subspace of the backward shift operator). I will present some recent work of Sarason and focus on the problem: When is a bounded operator on a model space a truncated Toeplitz operator?

Notes

February 12

Bill Ross, University of Richmond

Truncated Toeplitz operators on finite dimensional spaces

In the previous talk I introduced the problem: When is a bounded operator on a model space a truncated Toeplitz operator? In this talk I, in some joint work with Warren Wogen and Joe Cima, will present a complete solution to this problem (in terms of matrix representations) when the model space is finite dimensional.

Notes

February 19

Tom Kriete, UVa

Toeplitz-composition C*-algebras with several generators

Suppose S denotes the shift operator on the Hardy space H2 and let C*(S) denote the C*-algebra generated by S and the identity (otherwise known as the continuous Toeplitz algebra). An old but beautiful theorem of Coburn identifies the quotient of C*(S) by the ideal K of compact operators as the algebra of continuous functions on the unit circle. I'll describe a recent analogue of Coburn's theorem (joint work with Barbara MacCluer and Jennifer Moorhouse) in which C*(S) is replaced by the C*-algebra generated by S and a finite number of composition operators induced by linear fractional non-automorphisms of the disk having no boundary fixed point.

February 26

David Sherman, UVa

Locally inner automorphisms of operator algebras

The first half of this talk will be a survey of automorphisms of operator algebras, beginning with the simplest possible cases. (Some participants may be surprised to hear that "inner" and "outer" have meaning outside Hardy spaces.) Then I will introduce locally inner automorphisms, which have two branches of ancestors, one from group theory and one from operator algebras. I will focus on the subtle line between "locally inner" and "inner."

The relevant paper is here.

March 4

NO SEMINAR - SPRING BREAK

March 11

David Sherman, UVa

Diagonal sums of automorphism orbits

I will start by summarizing some results from the previous talk, adding more background and tying up loose ends. Then I will consider a very natural question about automorphism orbits in C*-algebras: does it make sense to form (diagonal) direct sums? Yes for B(H), but no in general. The precise answer involves locally inner automorphisms.

March 18

Chuck Akemann, UCSB

The Kadison-Singer problem and related questions

In a 1959 paper, Dick Kadison and Iz Singer raised questions about the relationships between the algebra B(H) of all bounded linear operators on a separable (infinite dimensional, complex) Hilbert space and its maximal commutative C*-subalgebras. While they answered some of these questions, they left open several tantalizing problems.

One such question, now known as the Kadison-Singer Problem, is as follows. If A is a maximal commutative C*-subalgebra of B(H) and f is a homomorphism of A into the complex numbers, does f have a unique, norm-preserving extension to all of B(H)? The algebra A of interest here is defined by taking an orthonormal basis for H and letting A be the algebra of all operators b in B(H) such that each element of the basis is an eigenvector for b.

At first glance this problem seems somewhat obscure because the homomorphisms for which uniqueness is in doubt only exist by means of the axiom of choice. However this question has been reduced to a question about finite matrices; in fact, there are many equivalent matrix versions of this question. In the talk I will describe one of the matrix versions that is easy to understand and relate it to some combinatorial questions that have already been solved (and some that have not). Despite the original form of the Kadison-Singer Problem, understanding this talk will only require a basic undergraduate background in complex linear algebra.

(Chuck will also give the department colloquium on Thursday, March 20th.)

March 25

NO SEMINAR

April 1

Katherine Heller, UVa

Adjoints of rationally induced composition operators

April 8

NO SEMINAR

April 15

Katie Quertermous, UVa

A lower bound on the essential norm of a difference of composition operators

April 22

Barbara MacCluer, UVa

Joining a compact operator to a non-compact operator by a continuous arc of composition operators on the Hardy space

April 29

Jan Spakula, Vanderbilt University

On uniform K-homology

Analytic K-homology (the dual homology theory to K-theory) of a non-compact space maps to K-theory of the Roe C*-algebra of this space via the coarse assembly map. The Roe C*-algebras of spaces reflect their coarse (large-scale) geometry. Whether the coarse assembly map is an isomorphism or not is the subject of the coarse Baum-Connes conjecture. This conjecture has applications in geometry, most notably it implies the Novikov conjecture on homotopy invariance of higher signatures.

In this talk, I will define a uniform version of analytic K-homology theory and prove a criterion for amenability of metric spaces in terms of their uniform K-homology. Furthermore, I will construct an assembly map from uniform K-homology of a metric space X to the K-theory of its uniform Roe C*-algebra. This is analogous to the classical (non-uniform) assembly map in the Coarse Baum-Connes conjecture.



You can reminisce about previous semesters at the links below:
Fall 2007 Spring 2007 Fall 2006 Spring 2006 Fall 2005 Spring 2005 Fall 2004 Spring 2004 Fall 2003 Spring 2003 Fall 2002

The seminar lost its senior participant, Bill Duren, in April.
Washington Post obituary
MAA obituary
Bill Duren and the Operator Theory Seminar -- short essay by Jim Rovnyak
A Career as a Scientific Generalist Based in Mathematics -- remarks delivered by Bill Duren to the UVa mathematics department on the occasion of his 100th birthday (Nov. 10, 2005)