Graph von Neumann algebras

In this paper, we will define a graph von Neumann algebra M-G over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra MG = M x alpha G, where G is the graph groupoid of G and alpha is the graph-representation. After defining a certain conditional expectation from M-G onto its M-diagonal subalgebra D-G, we can see that this crossed product algebra M-G is *-isomorphic to an amalgamated free product *D-Ge is an element of E(G) M-e, where M-e = vN (M x (alpha) G(e), D-G), where G(e) id the subset of G consisting of all reduced words in {e, e(-1)} and M x (alpha) G(e) is a W*-subalgebra of 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 M-G is isomorphic to a Banach space D-G circle plus (circle plus(w*is an element of E(G)R*) M-w*(o)), where E(G)(r)(*) is a certain subset of the set E(G)* of all words in the edge set E(G) of G.