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


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


September 18

Stephen Hardy, UVa

A short history and proof of Lin's theorem

Are almost commuting self-adjoint matrices near commuting self-adjoint matrices? This question, introduced in the 1960s and championed by mathematicians such as Halmos, was finally satisfactorily answered in 1994 by Huaxin Lin. We will briefly explore the history of this and related approximation problems, and sketch the elementary proof of this remarkable result due to Friis and Rordam.

September 25

Kevin Beanland, Virginia Commonwealth University

Constructions in Banach space theory

In 1974 Boris Tsirelson constructed the first infinite-dimensional Banach space that contained no "isomorphic copies" of lp for 1 ≤ p < ∞ or c0. This solved an old problem in Banach space theory. Perhaps more interesting though is the method he developed to carry out the construction. The so-called saturation method was inspired by forcing from logic and is now the predominant way to construct Banach spaces with prescribed properties. This method has led to numerous breakthroughs during the past 40 years, which I plan to discuss. In this talk I will describe this construction and explain its versatility. In the end I will state several open problems.

October 2

Scott Atkinson, UVa

A discussion of Grothendieck's Inequality

Grothendieck's Inequality is a celebrated result demonstrating a nontrivial relation between Hilbert space, L1, and L. Its interest reaches into several fields--to mention a few: functional analysis, data structures, optimization and control, and combinatorics. In this talk we will look at several formulations of the inequality--proving one.

October 4: Ed Effros (UCLA), UVa department colloquium Classical and Quantum Variables, and Quantized Functional Analysis
October 6-7: ECOAS conference, University of Tennessee

October 9

No meeting - reading day

October 16

Bill Ross, University of Richmond

Sturm-Liouville operators and deBranges-Rovnyak spaces I

Over the past several years I've been talking about models for symmetric operators with equal deficiency indices. By models I mean representing these symmetric operators as multiplication by the independent variable on either some Lebesgue type space of functions on the real line or some Hilbert space of analytic functions on C\R. Throughout the past several years, I have also discussed the Livsic characteristic functions for these operators and the fact that they are unitary invariants. In this talk, I plan to discuss deBranges-Rovnyak spaces as a model for these types of operators. The first part of these talks will be a survey of vector-valued deBranges-Rovnyak spaces while the second part of these talks will be about applying this deBranges-Rovnkak model to Sturm-Liouville differential operators. We will come full circle here in that the deBranges-Rovnyak model will give us some function theoretic information about the Livsic characteristic function. This is all joint work with Alexandru Aleman (Lund) and Rob Martin (University of Cape Town).

October 23

Bill Ross, University of Richmond

Sturm-Liouville operators and deBranges-Rovnyak spaces II

Over the past several years I've been talking about models for symmetric operators with equal deficiency indices. By models I mean representing these symmetric operators as multiplication by the independent variable on either some Lebesgue type space of functions on the real line or some Hilbert space of analytic functions on C\R. Throughout the past several years, I have also discussed the Livsic characteristic functions for these operators and the fact that they are unitary invariants. In this talk, I plan to discuss deBranges-Rovnyak spaces as a model for these types of operators. The first part of these talks will be a survey of vector-valued deBranges-Rovnyak spaces while the second part of these talks will be about applying this deBranges-Rovnkak model to Sturm-Liouville differential operators. We will come full circle here in that the deBranges-Rovnyak model will give us some function theoretic information about the Livsic characteristic function. This is all joint work with Alexandru Aleman (Lund) and Rob Martin (University of Cape Town).

October 30

Pavlos Motakis, National Technical University of Athens

Banach spaces of functions of bounded generalized variation

Here is the abstract.

November 1: Gilles Pisier (Paris VI and Texas A&M), UVa department colloquium
November 3: VOTCAM meeting, Virginia Commonwealth University

November 6

Katie Quertermous, James Madison University

Unitary equivalence of weighted composition operators on weighted Bergman spaces

Recently, several authors have studied the structures of C*-algebras generated by linear-fractionally-induced composition operators on the Hardy space of the disk. In this talk, we will extend some of these results to the weighted Bergman spaces by establishing a unitary equivalence between weighted composition operators of specific forms on the spaces.

November 13

Kate Juschenko, Vanderbilt University

Small spectral radius and percolation constants on non-amenable Cayley graphs

Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group Γ, does there exist a generating set S such that the Cayley graph (Γ,S), without loops and multiple edges, has non-unique percolation, i.e., pc(Γ,S) < pu(Γ,S)? We show that this is true if Γ contains an infinite normal subgroup N such that Γ/N is non-amenable. Moreover for any finitely generated group G containing Γ there exists a generating set S' of G such that pc(Γ,S') < pu(ΓS'). In particular this applies to free Burnside groups B(n,p) with n > 1, p > 664. We also explore how various non-amenability numerics, such as the isoperimetric constant and the spectral radius, behave on various growing generating sets in the group. Some application of the above results to group C*-algebras will be given. Joint with T. Nagnibeda-Smirnova.

November 20

No meeting - day before Thanksgiving Break

November 27

Ali Kavruk, University of Illinois

Nuclearity aspects of non-commutative cubes

The study of B.S. Tsirelson on relativistic and non-relativistic quantum measurements via self-adjoint operators acting on a Hilbert space goes back to 80's. Recently, on a research initiated by V.I. Paulsen, the operator systems arisen from group representations have been studied. Non-commutative cubes, at the intersection of both of these theories, have been proven to be at the heart of classical nuclearity theory of C*-algebras.

In this talk, after a brief introduction to operator systems, we will focus on non-commutative cubes and exhibit several formulations of their construction in terms of quotient, duality, coproducts and embeddability. We will see that a C*-algebra has Lance's weak expectation property if and only if its minimal and maximal tensor products with NC(k), the non-commutative cube of order k, coincide for some k > 2. Moreover, a C*-algebra is nuclear (in the sense of Lance) if and only if its minimal and maximal tensor products with the dual non-commuative cube, NC(k)*, coincide for some k>2. This establishes a non-commutative analogue of a classical result of Namioka and Phelps on function systems. Non-commutative cubes, exemplifying non-exact operator system with the lifting property, have also been shown to be a proper object to study several classical open problems in operator theory. We shall discuss an equivalent formulation of Connes' embedding problem (via Kirchberg's conjecture) in this low dimensional setting.

This talk is based on joint research with D. Farenick, I.G. Todorov and V.I. Paulsen.





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