A Notion of Singularity for von Neumann Algebras

The main result established in this thesis is the notion of property $(P)$, which extends the idea of singularity for amenable subgroups of countable discrete groups presented by Boutonnet and Carderi \cite{boca15}. While the result of Boutonnet and Carderi provides a means of proving maximal amenability for von Neumann algebras $L\Lambda \subset L\Gamma$ arising from singular subgroups, property $(P)$ generalizes the result to certain amenable von Neumann subalgebras $A \subset M$ which may or may not arise from groups. That is, property $(P)$ provides a dynamical condition for identifying such maximal amenable subalgebras. We then provide computations in the radial masa $A\subset L\mathbb{F}_K$, which take advantage of R\u{a}dulescu's basis $\ell^2\mathbb{F}_K \ominus A$ \cite{ra91} and mirror similar computations used by Cameron, Fang, Ravichandran, and White to demonstrate the maximal amenability of the radial masa \cite{cafarawh10}.