From graphs to free products

TL;DR

This paper constructs finite von Neumann algebras from weighted graphs, revealing their structure as free products with amalgamation, and providing natural examples of free probability distributions and operators.

Contribution

It introduces a novel graph-based construction of von Neumann algebras that generalizes free group factors and yields natural free probability models.

Findings

Von Neumann algebra of a flower graph is a free group factor.

The algebra is a free product with amalgamation over a finite-dimensional abelian algebra.

Provides models of free Poisson, circular, and semi-circular operators.

Abstract

We investigate a construction which associates a finite von Neumann algebra $M(\Gamma,\mu)$ to a finite weighted graph $(\Gamma,\mu)$. Pleasantly, but not surprisingly, the von Neumann algebra associated to to a `flower with $n$ petals' is the group von Neumann algebra of the free group on $n$ generators. In general, the algebra $M(\Gamma,\mu)$ is a free product, with amalgamation over a finite-dimensional abelian subalgebra corresponding to the vertex set, of algebras associated to subgraphs `with one edge' (or actually a pair of dual edges). This also yields `natural' examples of (i) a Fock-type model of an operator with a free Poisson distribution; and (ii) $\C \oplus \C$-valued circular and semi-circular operators.