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 G _e . 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.