Diagram Groupoids and von Neumann Algebras

Abstract

The main purpose of this paper is to study certain algebraic structures induced by directed graphs. We have studied graph groupoids, which are algebraic structures induced by given graphs. By defining a certain groupoid-homomorphism δ on the graph groupoid \({\mathbb{G}}\) of a given graph G, we define the diagram of G by the image \({\delta(\mathbb{G})}\) of δ, equipped with the inherited binary operation on \({\mathbb{G}}\). We study the fundamental properties of the diagram \({\delta(\mathbb{G})}\), and compare them with those of \({\mathbb{G}}\). Similar to Cho (Acta Appl Math 95:95–134, 2007), we construct the groupoid von Neumann algebra \({\mathcal{M}_{G}=vN(\delta(\mathbb{G}))}\), generated by \({\delta(\mathbb{G})}\), and consider the operator algebraic properties of \({\mathcal{M}_{G}}\). In particular, we show \({\mathcal{M}_{G}}\) is *-isomorphic to a von Neumann algebra generated by a family of idempotent operators and nilpotent operators, under suitable representations.