Speaker: Junhao Shen, University of New Hampshire

Date: Tu, March 3

Abstract: Let H be a separable complex Hilbert space, and let B(H) denote the set of bounded linear operators on H. The concept of reducible operators was introduced by P. Halmos in 1968: an operator T in B(H) is reducible if it has nontrivial reducing closed subspaces in H. In his famous list of “Ten Problems in Hilbert Space,” he asked whether every bounded linear operator on H is a norm limit of reducible operators. It is not hard to see that on a finite-dimensional Hilbert space, the answer is negative. In the infinite-dimensional space case, this question was answered affirmatively by D. Voiculescu using his celebrated non‑commutative Weyl–von Neumann theorem. A von Neumann algebra is a self-adjoint subalgebra of B(H) that is closed in the weak operator topology and contains the identity of B(H). A factor is a von Neumann algebra whose center consists only of scalar multiples of the identity. Factors were further classified by Murray and von Neumann into type I, II, and III factors. By definition, B(H) is a type I factor. In this talk, we will discuss the concept of reducible operators in factors and study the norm‑density properties of reducible operators in various types of factors by extending Voiculescu’s non‑commutative Weyl–von Neumann theorem to the setting of von Neumann algebras. More specifically, we will show that the set of reducible operators is norm dense in properly infinite factors, while this need not be the case in finite factors.