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

TL;DR

The paper resolves the level-3 instance of the two-sided GJS construction for the group planar algebra of a finite group. It identifies a distinguished generator $Z$ in $P_{(5,+)}$, proves that $Z$ and $P_{(3,+)}$ generate the dense algebra $A_G^3$ and are free, and shows $Z$ is a free Poisson element whose von Neumann algebra is known. This yields a free-product decomposition $M^3 = vN(Z)*P_{(3,+)}$, which, together with the Dykema–Radulescu free-product formula, evaluates to the interpolated free group factor $LF(1+2(n-1)/n^2)$ with $n=|\,\mathcal{G}\,|$, making $M^3$ a matrix amplification of $M^1$. The approach keeps combinatorial control via planar diagrams and free probability, clarifying the level-3 structure and suggesting a path to higher levels through iterative Jones basic constructions; the result depends only on $n$ and not on the isomorphism type of the group.

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.

Figures (12)

• Figure 1: A typical element of $A_m^k$.
• Figure 2: Two multiplication operations on $A^k$
• Figure 3: Obtaining $a^{\dagger}$ from $a$: apply the planar algebra adjoint and then the rotation tangle.
• Figure 4: Diagrammatic formula for the trace $\tau_G$ on $A_G^k$.
• Figure 5: The self-adjoint element $Z\in P_{(5,+)}$.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Lemma 3
• Theorem 4
• Theorem 5
• Proposition 6
• proof
• Proposition 7
• proof
• Lemma 8