Date of Award

Spring 2026

Abstract

The study of reducible and irreducible operators in von Neumann algebras has long been a central topic in operator theory. In this dissertation, we obtain results in two directions.

A classical question of Paul Halmos asks whether every operator on a separable infinite-dimensional Hilbert space is a norm limit of reducible operators. Junhao Shen and Shi investigated this problem for non-Gamma type II₁ factors and showed that the set of reducible operators is nowhere dense in such algebras. In Chapter 2, we extend this result to non-Gamma pairs of type II₁ factors: for a non-Gamma inclusion N ⊆ M, the relatively reducible operators in M are nowhere dense near N.

The study of the density of irreducible operators dates back to Paul Halmos (1968), who proved that irreducible operators are norm dense in B(H). Later, Domingo Herrero (1989) established the corresponding density result in the Schatten p-norm for p > 1. However, the case p = 1 remains open, as the main techniques in Herrero’s work do not apply. Specifically, given an operator T = A + iB in B(H) and ε > 0, it is unknown whether there exists a trace-class operator K with ‖K‖₁ < ε such that T + K is irreducible. Fang and collaborators gave an affirmative answer for a large class of operators. In Chapter 3, we continue this line of investigation by considering the following cases: when C*(A,B) has nontrivial center; when W*(A,B) is finite or semifinite with nontrivial center; and when W*(A,B) is a type II₁ factor with nonzero projections P,Q ∈ W*(A) such that PBQ = 0.

Document Type

Dissertation

First Advisor

Junhao Shen

Second Advisor

Junhao Shen

Third Advisor

Donald Hadwin

Department or Program

Mathematics

Degree Name

Doctor of Philosophy

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