Proceedings - Mathematical Sciences

, Volume 122, Issue 4, pp 547–560 | Cite as

From graphs to free products

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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) \({\mathbb C} \oplus {\mathbb C}\)-valued circular and semi-circular operators.

Keywords

von Neumann algebras associated to graphs Guionnet–Jones–Shlya-khtenko construction (for possibly non-bipartite graphs) free Poisson variable 
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Notes

Acknowledgements

We would like to thank M Krishna for patiently leading us through the computation of Cauchy transforms of rank-one perturbations as we struggled with an apparent contradiction, which was finally resolved when we realised a problematic minus sign stemming from a small mistake in choice of square roots. (We claim no originality for this problem, for the same incorrect sign also surfaces on page 33 of [9] – cf. our formula (2.2) and the formula there for g, when − ∞ < t < − 2.)

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Copyright information

© Indian Academy of Sciences 2012

Authors and Affiliations

  1. 1.The Institute of Mathematical Sciences, CIT CampusChennaiIndia

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