Date of Award
Fall 2024
Project Type
Dissertation
Program or Major
Mathematics
Degree Name
Doctor of Philosophy
First Advisor
JUNHAO SHEN
Second Advisor
MARIANNA SHUBOV
Third Advisor
DONALD HADWIN
Abstract
Let $\mathcal{H}$ be a complex Hilbert space, and let $\mathcal{B(H)}$ be the set of all bounded linear operators on $\mathcal{H}$. An operator $T\in\mathcal{B(H)}$ is said to be reducible if there exists a nontrivial projection that commutes with $T$. Otherwise, it is said to be an irreducible operator in $\mathcal{B(H)}$. Halmos showed that the set of irreducible operators is an operator norm-dense subset in $\mathcal{B(H)}$ and raised the question of whether the set of reducible operators is an operator norm-dense subset in $\mathcal{B(H)}$. In this dissertation, we extend the notion of reducible and irreducible operators in the von Neumann algebra setting while considering both separable and non-separable cases. Next, we review the most recent advancements related to the operator norm density of the set of reducible and the set of irreducible operators in factors. We then extend these results into von Neumann algebras using the theory of direct integration. Recently, Shen and Shi showed that any operator in a separable factor is the sum of two irreducible ones. We further extend this result to von Neumann algebras.\\
An operator $T$ in a von Neumann algebra $\mathcal{M}$ is said to be a generator if the von Neumann algebra generated by $T$ is $\mathcal{M}$. A famous question of Kadison asks whether any separable von Neumann algebra is singly generated. We look at some of the most recent developments related to Kadison's question and show that if a separable factor is singly generated, the set of single generators is topologically large in the operator norm topology. Further, we extend the result for von Neumann algebras using the theory of direct integration.\\
For $i=1,2$, assume $\tau_i$ is a semi-finite normal trace on the von Neumann algebra $\mathcal{M}_i$. Given an isometry $T$ between two non-commutative $L_p$ spaces $L_p(\mathcal{M}_1,\tau_1)$ and $L_p(\mathcal{M}_2,\tau_2)$, F. J. Yeadon showed that there exists, uniquely, a partial isometry $W$, an unbounded positive self-adjoint operator $B$ both in $L_p(\mathcal{M}_2,\tau_2)$ such that $T(X)=WBJ(X) \text{ for all } X\in L^p(\mathcal{M}_1,\tau_1)\cap \mathcal{M}_1$. Define $\mathcal{L}_\alpha(\mathcal{M},\tau)$ to be the completion of $\mathfrak{F}=\{A\in\mathcal{M}:\tau(S(A)<\infty\}$ (here $S(A)$ is the support projection of $A$) with respect to the norm $\|A\|_\alpha=\frac 1 n \sum_{k=1}^n \tau(|A|^{2k})^{\frac{1}{2k}}$. In the last part of this dissertation, we extend F. J. Yeadon's result for $\mathcal{L}_\alpha(\mathcal{M},\tau)$ space.
Recommended Citation
Adappa, Sukitha, "Reducible Operators, Irreducible Operators, and Generators in von Neumann Algebras" (2024). Doctoral Dissertations. 2851.
https://scholars.unh.edu/dissertation/2851