Seminar in operator theory and operator algebras (MATH 9310)
Spring 2012

The seminar is organized by David Sherman. We meet Tuesdays 3:30-4:30 in Kerchof 326.

January 31

Tom Kriete, UVa

Direct integral basics

Direct integral theory provides a useful notion of continuous direct sum for Hilbert spaces. In three talks I will present the basic construction in detail, along with the classical application to multiplicity theory for spectral measures.

February 7

No meeting

February 14

Tom Kriete, UVa

Direct integral basics

February 21

Tom Kriete, UVa

Direct integral basics

February 28

No meeting

March 6


March 13

No meeting

March 20

Leonel Robert, University of Louisiana at Lafayette

The cone of lower semicontinuous traces on a C*-algebra

A trace is a linear map from the positive elements of a C*-algebra to the extended positive real numbers satisfying the ``trace identity". The lower semicontinuous traces of a C*-algebra are suitable non-commutative analogues of the sigma-additive positive measures on the Borel sigma-algebra of a topological space. Traces have been the subject of study since the early days of C*-algebra theory. The cone of all lower semicontinuous traces, however, has been less studied. This object is of special interest in the classification of C*-algebras and related questions. I will talk about the cone of lower semi-continuous traces of a C*-algebra; what I know and what I do not know about it.

Ed Effros (UCLA) will be giving the department colloquium on March 22. (Canceled due to illness.)

March 27

No meeting

Roger Smith (Texas A&M) will be giving the department colloquium on March 29.

April 3

No meeting

April 10

Craig Kleski, UVa

Graph C*-algebras

A directed graph may be represented as operators on a Hilbert space: the vertices are mutually orthogonal projections, and the edges are operators between subspaces determined by the projections. The algebra of operators generated by this set is a graph C*-algebra. Algebraic properties of such C*-algebras are related to properties of the underlying graph. These algebras have been intensely studied in the past 15 years. In this talk, we will introduce the basics of graph C*-algebras and, time permitting, discuss generalizations.

April 17

Vrej Zarikian, US Naval Academy

The unique pseudo-expectation property, and a new proof of Mercer's extension theorem

Click here for the abstract.

Click here for the slides.

April 24

Jon Bannon, Siena College

Correspondences of finite von Neumann algebras and joinings of measurable dynamical systems

A *joining* of two measure-preserving dynamical systems is a (diagonally) invariant measure on the product of the underlying measurable spaces whose marginals are the original invariant measures. Over the past 30 years joining theory has become an important part of ergodic theory, and has been used to generate many striking counterexamples to old questions, as well as elegant proofs of classical results. If the acting group is trivial, a joining of two probability spaces determines and is determined by a binormal bimodule, or correspondence, of the associated von Neumann algebras of essentially bounded measurable functions. Correspondences play a role in many of the fundamental structural results for finite von Neumann algebras. In this setting, joinings generalize naturally to equivariant correspondences, which have appeared recently in the guise of bimodular completely positive maps in Popa's deformation/rigidity strategy. In this talk we discuss the above generalization of joining theory, presenting basic examples of joinings and proving some results motivated by the analogy between the two theories.

May 1

Mor Katz, UVa

Composition operators on Hardy spaces of a half plane

Although corresponding Hardy spaces of the disc and half-plane are isomorphic, composition operators act very differently in the two cases. Valentin Matache showed that a composition operator is bounded on the Hardy space of the half-plane if and only if its symbol has finite angular derivative at infinity, and that there are no compact composition operators on the half-plane. Sam Elliott and Michael Jury gave a simplified proof and strengthened this result. In this talk we will see the simplified proof and show that the norm, essential norm and spectral radius of a composition operator on the half-plane are all equal to the angular derivative of the symbol at infinity.

You can reminisce about previous semesters at the links below:
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