This dissertation consists of three papers I have written or helped write during my time at UC Berkeley. The three papers all fall under the common theme of exploring connections between free probability and planar algebras.
In Chapter 2, an amalgamated free product algebra, $\cN(\Gamma)$, is constructed for any connected, weighted, loopless graph $\Gamma$, and its isomorphism class can be nicely read off based on the weighting data on the graph. This construction was heavily influenced by an algebra appearing in in work of Guionnet, Jones and Shlyakhtenko, and it is used, along with some standard embedding arguments, to show that the factors of Guionnet, Jones, and Shlyakhtenko appearing in are isomorphic to $L(F_{\infty})$ when the planar algebra is infinite depth.
Chapter 3 is the paper ``Rigid $C^{*}-$tensor categories of bimodules over interpolated free group factors" which was co-authored with Arnaud Brothier and David Penneys. In this paper, we establish an unshaded planar algebra structure (with multiple colors of strings) which can be used to model a countably generated rigid $C^{*}-$tensor category, $C$. We use this to construct a category of bifinite bimodules over a $II_{1}$ factor $M_{0}$, and we show that this category is equivalent to $C$. Finally, we use the work in Chapter \ref{chap:graph} to show that the factor $M_{0}$ is isomorphic to $L(F_{\infty})$.
Chapter 4 is a note regarding multishaded planar algebras. The problem studied consists of placing the ``Fuss-Catalan" potential on a specific kind of subfactor planar algebra $Q$, which is the standard invariant for and inclusion $N \subset M$ with an intermediate subfactor $P$. This potential is best understood by augmenting $Q$, forming a bigger multishaded planar algebra $\mathcal{P}$. The isomorphism classes of the algebras $M_{\alpha}$ associated to $\mathcal{P}$ will be computed explicitly. While the isomorphism class of the von Neumann algebras $N_{k}^{\pm}$ associated to $Q$ are still not yet known, they will be shown to be contained in a free group factors and contain a free group factors. This potential is shown to yield a nice free product expression for the law of $\cup$, an element which plays a critical role in understanding algebras that arise from this construction. Many of the ideas in this chapter influenced the work in the Chapter 3.
