Free Probability, Planar Algebras, and the Multicolored Guionnet-Jones-Shlyakhtenko Construction

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