# Free transport for interpolated free group factors

## TL;DR

This paper extends free transport techniques to interpolated free group factors using a graph-based von Neumann algebra model, operator-valued free difference quotients, and Schwinger-Dyson equations.

## Contribution

It introduces a new free transport framework for interpolated free group factors based on graph models and operator-valued calculus, expanding prior work on free group factors.

## Key findings

- Constructed a von Neumann algebra model from weighted graphs
- Developed an operator-valued Schwinger-Dyson equation for generators
- Established free transport for perturbations of the equation

## Abstract

In this article, we study a form of free transport for the interpolated free group factors, extending the work of Guionnet and Shlyakhtenko for the usual free group factors. Our model for the interpolated free group factors comes from a canonical finite von Neumann algebra $\mathcal{M}(\Gamma, \mu)$ associated to a finite, connected, weighted graph $(\Gamma,V,E, \mu)$. With this model, we use an operator-valued version of Voiculescu's free difference quotient to state a Schwinger-Dyson equation which is valid for the generators of $\mathcal{M}(\Gamma, \mu)$. We construct free transport for appropriate perturbations of this equation. Also, $\mathcal{M}(\Gamma, \mu)$ can be constructed using the machinery of Shlyakhtenko's operator-valued semicircular systems