Seminar in Operator Theory and Operator Algebras - Fall 2010, University of Virginia

Seminar in operator theory and operator algebras (MATH 9310)
Fall 2010

This semester the seminar will run somewhat more irregularly than usual. Our standard meeting time is Tuesday 3:30-4:30 in Kerchof 326.

September 14	Andreas Hartmann, Université Bordeaux I (visiting University of Richmond) Interpolation and weak interpolation in backward shift invariant subspaces
September 28	James Rovnyak, UVa Schur complements and positive polynomials in one and several variables (expository lecture)
October 5	Craig Kleski, UVa Commutative and noncommutative Shilov boundaries The commutative Shilov boundary is a generalization of the maximum modulus theorem to uniform algebras in C(X). Since C(X) is the prototypical abelian C-algebra, this boundary has a natural generalization to certain subspaces and subalgebras of (noncommutative) C-algebras. It provides an answer to the following question: how is the C-algebra generated by a concrete operator system S related to the C-algebra generated by a completely isometric image of S?
October 19	Bill Ross, University of Richmond Algebras of truncated Toeplitz operators I Recently, Nic Sedlock characterized the maximal algebras in the truncated Toeplitz operators. In these two talks, we will give an exposition of Sedlock's work and discuss when different maximal algebras of truncated Toeplitz operators are spatially isomorphic to each other.
October 26	Bill Ross, University of Richmond Algebras of truncated Toeplitz operators II See previous week's abstract.
November 2	Tom Kriete, UVa Distance in the Calkin algebra between composition operators
November 16	Andreas Hartmann, Université Bordeaux I (visiting University of Richmond) Analytic continuation in backward shift invariant subspaces
November 30	David Sherman, UVa A unified representation theory for Hilbert space operators I We prove general results that hold in any of four categories of unital singly-generated objects: C*-algebras, hereditary manifolds, operator algebras, and operator spaces. An overarching goal is to give various answers to the following question: for x and y operators on separable Hilbert spaces, when is there a morphism from the object generated by x to the object generated by y, taking x to y? In this first talk I will explain most of the main concepts. Then I'll prove that any morphism can be written as the composition of an amplification, an approximate unitary equivalence, and a spatial morphism. (Historical note: some of this is directly descended from ideas that were first exposed in a series of UVa operator seminars given by Jim Agler in the 1980s.)
December 7	David Sherman, UVa A unified representation theory for Hilbert space operators II We prove general results that hold in any of four categories of unital singly-generated objects: C-algebras, hereditary manifolds, operator algebras, and operator spaces. An overarching goal is to give various answers to the following question: for x and y operators on separable Hilbert spaces, when is there a morphism from the object generated by x to the object generated by y, taking x to y? In this second talk I will explain how each category is paired with a specific operator topology, and I'll introduce the apparently new semistrong* operator topology. Then I'll generalize a theorem of Hadwin relating closed unitary orbits to spatial representations. Putting this together with last week's theorem, we get that the set of representations of x is exactly the closure of {x} in an appropriate topology.

You can reminisce about previous semesters at the links below:
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