Graph von Neumann Algebras

Abstract

In this paper, we will define a graph von Neumann algebra \(\Bbb{M}_{G}\) over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra \(\Bbb{M}_{G}\) = M ×α\(\Bbb{G}\) , where \(\Bbb{G}\) is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from \(\Bbb{M}_{G}\) onto its M-diagonal subalgebra \(\Bbb{D}_{G},\) we can see that this crossed product algebra \(\Bbb{M}_{G}\) is *-isomorphic to an amalgamated free product \(\underset{e\in E(G)}{\,*_{\Bbb{D}_{G}}}\)\(\Bbb{M} _{e},\) where \(\Bbb{M}_{e}\) = vN(M ×α\(\Bbb{G}_{e},\)\(\Bbb{D}_{G}),\) where \(\Bbb{G}_{e}\) is the subset of \(\Bbb{G}\) consisting of all reduced words in {e, e–1} and M ×α\(\Bbb{G} _{e}\) is a W*-subalgebra of \(\Bbb{M}_{G},\) as a new graph von Neumann algebra induced by a graph Ge. Also, we will show that, as a Banach space, a graph von Neumann algebra \(\Bbb{M}_{G}\) is isomorphic to a Banach space \(\Bbb{D}_{G}\) ⊕ \((\underset{w^{*}\in E(G)_{r}^{*}}{\oplus }\)\(\Bbb{M}_{w^{*}}^{o}),\) where \(E{\left( G \right)}^{*}_{r}\) is a certain subset of the set E(G)* of all words in the edge set E(G) of G.