On Maximal Amenable Subalgebras of Amalgamated Free Product von Neumann Algebras

In this thesis, we establish a sufficient condition for an amenable von Neumann algebra to be a maximal amenable subalgebra of an amalgamated free product von Neumann algebra. In particular, if $P$ is a diffuse maximal amenable von Neumann subalgebra of a finite von Neumann algebra $N_1$, and $B$ is a von Neumann subalgebra of $N_1$ with the property that no corner of $P$ embeds into $B$ inside $N_1$ in the sense of Popa's intertwining by bimodules, then we conclude that $P$ is a maximal amenable subalgebra of the amalgamated free product of $N_1$ and $N_2$ over $B$, where $N_2$ is another finite von Neumann algebra containing $B$. To this end, we utilize Popa's asymptotic orthogonality property. We also observe several special cases in which this intertwining condition holds, and we note a connection to the Pimsner-Popa index in the case when we take $P=N_1$ to be amenable