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arXiv · 1102.4413

From graphs to free products

Madhushree Basu·Vijay Kodiyalam·V. S. Sunder
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Abstract

We investigate a construction 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 to a `flower with $n$ petals' is the group von 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) $\C \oplus \C$-valued circular and semi-circular operators.

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Madhushree Basu, Vijay Kodiyalam, V. S. Sunder. 2011-02-22. From graphs to free products. https://arxiv.org/abs/1102.4413

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