From graphs to free products

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.

14 pages, 1 figure

References in corpus (1)

• Random matrices, free probability, planar algebras and subfactors