Colloquia: Invariant subspaces and decompositions of operators in finite von Neumann algebras

Abstract: When we want to figure out a matrix A ∈ Mn(C), its eigenvectors are the key. We would like to treat similarly operators T on an infinite dimensional Hilbert space. Here, invariant subspaces of T would be of great interest. Unfortunately, we don’t know if they exist, in general. The situation is somewhat better if the von Neumann algebra generated by T has a faithful, bounded, trace (which is akin to the usual matrix trace on Mn(C)). Then we say that T belongs to a finite von Neumann algebra. We will describe various results about invariant subspaces and decompositions for such operators. These include analogues of Schur’s upper triangular forms and the Jordan canonical form.

Host: John McCarthy

Reception at 3:30 in Cupples I, Room 200 (Lounge).