Seminar in Operator Theory and Operator Algebras - Spring 2009, University of Virginia

Seminar in operator theory and operator algebras (MATH 931)
Spring 2009

The seminar is organized by Tom Kriete, and this webpage is maintained by David Sherman. We meet Tuesdays 3:30-4:30 in Kerchof 326.

January 20	Samuel Krushkal, Bar Ilan University (visiting UVa) Inversion of Ahlfors and Grunsky inequalities The classical Ahlfors inequalty (of 1952) 1/ρ_L ≤ q_L for closed oriented Jordan curves L on the sphere is fundamental in the theory of Fredholm eigenvalues; here q_L is the reflection coefficient of L and ρ_L is its first nontrivial eigenvalue. It is intrinsically connected with the strengthened Grunsky inequality for the Grunsky and Teichmüller norms of functions with quasiconformal extensions. The long standing problem of inversion of these inequalities was solved only recently by applying geometric methods involving the complex geometry of universal Teichmüller space (space of deformation of conformal structures) and the curvature properties of general Finsler metrics. I will briefly outline the main ideas and arguments.
January 27	(no meeting)
February 3	Bill Ross, University of Richmond The inverse Jordan problem for truncated Toeplitz operators Is every n×n matrix similar to a Toeplitz matrix? In general the answer is no. However, every n×n matrix is similar to a truncated Toeplitz operator.
February 10	Craig Kleski, UVa A survey of the representation theory of C-algebras In this talk, I will discuss the representation theory for various classes of C-algebras. In this context, we will associate to each C*-algebra two important objects -- the spectrum and the primitive ideal space. The degree to which these objects agree has interesting implications regarding representations.
February 17	Lisa Clark, Susquehanna University Groupoid C-algebras One method operator algebraists developed to better understand C-algebras is to build them from simpler objects. For example, we can construct a C-algebra from a group, a group action, and a directed graph. In all three of these cases, properties of the C-algebra directly correspond to properties of the object from which they were built. In this talk, I will describe how to build a C-algebra from a groupoid. Groupoid C-algebras are generalizations of group, group action, and graph algebras. I will start out by defining what is meant by a groupoid, giving a number of examples. Next I will describe the process of building a C-algebra from a groupoid. Finally, time permitting, I will show some results in which topological properties of the groupoid characterize the spectrum of the associated groupoid C-algebra.
February 24	Craig Kleski, UVa A survey of the representation theory of C-algebras, part 2 Let A be a C-algebra. For the second part of this survey, we'll examine in some detail the topological structure of Prim(A) and Â. This will lead to a discussion of continuous trace C*-algebras.
March 3	SPRING BREAK
March 10	Lon Mitchell, Virginia Commonwealth University C-algebras generated by isometries We will define and explore both old and new results on C-algebras generated by isometries, which include the Cuntz algebras and so-called graph algebras. In particular, we will mention some of the recent work by Kirchberg that has used Cuntz algebras to characterize separable nuclear C*-algebras.
March 17	Matthew Neal (UVa Ph.D. '99), Denison University Jordan structures in Banach space theory and operator space theory In the sixties, Douglas proved that an isometric copy of L¹ in a second L¹ space is always contractively complemented. In the eighties, Kirchberg generalized this result to von Neumann algebras, showing that an isometric copy of the predual of a von Neumann algebra inside the predual of a second von Neumann algebra is always contractively complemented. Thus, preduals of von Neumann algebras are said to be projectively rigid. However, unlike L¹ spaces they are not projectively stable, i.e. the range of a normal contractive projection on a von Neumann algebra is not isometric to another von Neumann algebra but is isometric to a more general structure called a JBW-triple. JBW-triples are of independent interest as exactly those dual Banach spaces whose unit ball has a transitive biholomorphic automorphism group. They are known to be contractively stable. In this talk we will completely solve the above projective rigidity problem for JBW-triples. We will also review the long history of these types of problems involving many authors in many different settings. In the second part of the talk we will focus on results in operator space theory that use JBW-triple theory. We give results about the operator space structure of the range of a contractive projection on B(H). In particular, we give a classification of Hilbert spaces that arise in this way. We will show the first known operator space classification of unital operator spaces. We also give an operator space classification of ternary rings of operators, which are fundamental invariants in the theory. This is joint work with David Blecher, Bernard Russo, and Eric Ricard.
March 24	David Sherman, UVa Ultraproducts in analysis I'll give an introduction to at least two flavors one encounters in functional analysis.
March 31	David Sherman, UVa Ultraproducts in analysis and model theory This is a continuation of the previous week's talk. I'll start with some discussion of the tracial ultraproduct construction for operator algebras, which is an important variation on the Banach space ultraproduct introduced last week. Neither of these ultraproducts is the same as the "classical" ultraproduct of model theory, and powerful theorems from logic do not directly apply to them. Then I'll sketch a wonderful hybrid approach which makes versions of the logical theorems available in the analytic context. This will give the audience some extra background for Ward Henson's colloquium on April 2.
April 7	Ugur Gul, visiting UVa A C-algebra generated by Fourier multipliers and Toeplitz operators, and its application to composition operators By the help of the Paley-Wiener theorem we construct a C-algebra generated by Fourier multipliers and Toeplitz operators on the Hardy space of the upper half-plane. And by the help of Cauchy integral formula we observe that a certain class of composition operators fall in this C*-algebra. We also transfer the results to the Hardy space of the unit disc.
April 14	Barbara MacCluer, UVa Aleksandrov measures, composition operators and compact differences This will be an expository talk, surveying some ways in which Aleksandrov measures have been used in composition operator questions. We will focus particularly on the question of describing when a difference of composition operators is compact.
April 21 3:30 PM Wilson 402	Distinguished lecture series Richard Kadison, University of Pennsylvania The early development of the theory of operator algebras
April 22 3:30 PM Kerchof 317	Distinguished lecture series Richard Kadison, University of Pennsylvania Operator algebras - a sampler
April 24 2:30 PM Kerchof 317	Distinguished lecture series Richard Kadison, University of Pennsylvania The Pythagorean theorem - a closer look
April 28	Katie Quertermous, UVa An introduction to crossed products of C-algebras In this expository talk, we will explore the crossed products of C-algebras by locally compact topological groups. We will begin by reviewing basic facts about topological groups and by looking at the group C*-algebra to motivate the crossed product construction. We will then build the crossed product and consider a few basic examples.

You can reminisce about previous semesters at the links below:
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