June 17, 2024
Thematic Program on Operator Algebras
July-December, 2007

Distinguished Lecture Series
November 6,7,8 -- 2007
Fields Institute

Free probability and the invariant subspace problem for von Neumann algebras

Does every operator T on a separable Hilbert space H 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: Let M be a von Neumann algebra 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 the orthogonal projection on K belongs to M).

In the special case that M is a II1-factor (i.e. a infinite dimensional von Neumann factor with a 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 in M for which L. G. Brown’s spectral distribution measure for T is not concentrated at a single point of the complex plane. The result is obtained in collaboration with Hanne Schultz (preprint 2006), and it relies in a crucial way on Voiculescu’s free probability theory. This result was previously known only for special operators such as Voiculescu’s circular operator and more generally for the DT-operators introduced in a paper in the American Journal of Mathematics (2004) in collaboration with Ken Dykema.

There will be three talks on this subject. The first will be a survey talk on this topic. In the second talk, I will go more into the technical details, and in particular explain how free probability enters in the proof of the main result. In the third talk, I will focus on examples and applications.

Haagerup's research is in operator theory, and covers many subareas in the subject which are currently very active - random matrices, free probability, C*-algebras & applications to mathematical physics. He is among the most significant contributors to the theory at present and his research has critical impact in virtually all areas of operator theory.

Haagerup is a professor at Odense University in Denmark. He has been an organizer of many major conferences, including several at Oberwolfach, MSRI, the Institut Mittag-Leffler and elsewhere. He is a member of the Royal Danish Academy of Science and Letters and of the Norwegian Academy of Science, and has been the Editor-in-Chief of Acta Mathematica.

Speakers in the Distinguished Lecture Series (DLS) have made outstanding contributions to their field of mathematics. The DLS consists of a series of three one-hour lectures.

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