On the two-sided Guionnet-Jones-Shlyakhtenko construction at level three

Abstract

We study the two-sided Guionnet-Jones-Shlyakhtenko construction applied to the group planar algebra $P(\mathcal{G})$ of a finite non-trivial group $\mathcal{G}$. This produces a sequence of von Neumann algebras $M^k$ for $k \geq 0$ with no natural inclusions. Focusing on level $k=3$, we show that the resulting von Neumann algebra $M^3$ is isomorphic to the interpolated free group factor LF$\left({1+\frac{2(n-1)}{n^2}}\right)$, where $n=|\mathcal{G}|$. Our approach keeps the combinatorics explicit and relies on standard tools from free probability and planar algebras.