Department of Mathematical Sciences
901.678.2482 		 

Functional Analysis
and Operator Theory
Colloquium

October 10, 2007
2:30 - 3:30 p.m.
Room 227, Dunn Hall
The invariant subspace problem
for von Neumann algebras		 UFFE HAAGERUP
University of Southern Denmark
Abstract:

Does every operator T on a Hilbert space have a non-trivial closed invariant subspace?
This is the famous and still open "invariant subspace problem" for operators on a Hilbert space.
A natural generalization of the problem is the following: Given a von Neumann algebra M on a Hilbert
space H, does every operator T in M have a non-trivial closed invariant subspace K affiliated with M?
(K is affiliated with M if and only if the orthogonal projection on K belongs to M.)
In the special case when M is a II_1 factor (an infinite dimension von Neumann factor with bounded trace),
it turns out that "almost all" operators in M have non-trivial closed invariant subspaces affiliated with M.
More precisely, this holds for all operators T in M for which L. G. Brown's spectral distribution measure
for T is not concentrated on a single point of the complex plane. The result is obtained in collaboration
with Hanne Schulz (2006), and it relies in a crucial way on Voiculescu's free probability theory.
Uffe Haagerup's research covers many areas in functional analysis and operator theory, including C*-algebras, von Neumann algebras, free probability, and random matrices. He is one of the most significant contributors to the theory at present, and his research has critical impact in virtually all areas of operator theory. He is a member of the Royal Danish Academy of Science and Letters and of the Norwegian Academy of Science, and he has served as Editor-in-Chief of Acta Mathematica.