[Turkmath:8060] Istanbul Bilgi University Departmental Seminar Dec 13

[Uğur Doğan]

Dear All,

Mohan Ravichandran (Sabanci University) will give a talk at our
departmental seminar on Tuesday (Dec 13) at 16:00 in the room D-135. The
title and abstract below (it may be more convenient to read the abstract in
the attached file).

Best,

Piotr Kowalski
------------------------------**------------------------------**------------
Title: The Schur-Horn theorem for von Neumann algebras

Abstract:

In a couple of delightful papers in 2002, Richard Kadison showed that
thinking about the Pythagorean theorem, or rather its converse, the so
called carpenter rule: If $a^2 + b^2 = c^2$, then there is a right-angled
triangle with sides $a, b$ and $c$, leads one to some fascinating problems
in functional analysis.

An intermediate step is the Schur-Horn theorem, which gives a complete
description of the relation between the diagonal and the eigenvalue list of
a Hermitian matrix. It says the following - The diagonal vector lies in the
simplex generated by orbit of the vector of eigenvalues under the action of
the Symmetric group. The theorem has found applications in a variety of
areas including optimization theory, convex analysis and symplectic
geometry.

Kadison's approach led him beyond Schur-Horn for matrices to two natural
infinite dimensional settings. The first is the algebra of all bounded
linear operators on a Hilbert space, $\mathcal{B}(\mathcal{H})$ and the
other is the context of finite von Neumann algebras Finite von Neumann
algebras are the natural setting for non-commutative measure theory.

The first case, for $\mathcal{B}(\mathcal{H})$, was very satisfactorily
analysed by Kadison, partly in joint work with William Arveson.
Establishing Schur-Horn for finite von Neumann algebras was left open. In
recent work, joint with Rajarama Bhat of ISI Bangalore, we solved this
problem completely. I will report on this work and explain applications of
our result to a few problems in the geometry of von Neumann algebras and
frame theory, among others.

The talk will be self-contained. Further, the first half of the talk will
be accesible to anybody who knows basic linear algebra.
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