From graphs to free products

Abstract

We investigate a construction (from Kodiyalam Vijay and Sunder V S, J. Funct. Anal. 260 (2011) 2635–2673) which associates a finite von Neumann algebra M(Γ,μ) to a finite weighted graph (Γ,μ). Pleasantly, but not surprisingly, the von Neumann algebra associated to a ‘flower with n petals’ is the group on Neumann algebra of the free group on n generators. In general, the algebra M(Γ,μ) 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⊕C-valued circular and semi-circular operators.