Characterization of Amalgamated Free Blocks of a Graph von Neumann Algebra

Abstract.

We identify two noncommutative structures naturally associated with countable directed graphs. They are formulated in the language of operators on Hilbert spaces. If G is a countable directed graphs with its vertex set V(G) and its edge set E(G), then we associate partial isometries to the edges in E(G) and projections to the vertices in V(G). We construct a corresponding von Neumann algebra ${\mathbb{M}}_G$ as a groupoid crossed product algebra $M \times_{\alpha} {\mathbb{G}}$ of an arbitrary fixed von Neumann algebra M and the graph groupoid ${\mathbb{G}}$ induced by G, via a graph-representation (or a groupoid action) α. Graph groupoids are well-determined (categorial) groupoids. The graph groupoid ${\mathbb{G}}$ of G has its binary operation, called admissibility. This ${\mathbb{G}}$ has concrete local parts ${\mathbb{G}}_e$ , for all e ∈ E(G). We characterize $M_{e} = M \times_{\alpha} {\mathbb{G}}_e$ of ${\mathbb{M}}_G$ , induced by the local parts ${\mathbb{G}}_e$ of ${\mathbb{G}}$ , for all e ∈ E(G). We then characterize all amalgamated free blocks ${\mathbb{M}}_{e} = \nu N(M_{e}, {\mathbb{D}}_G)$ of ${\mathbb{M}}_G$ . They are chracterized by well-known von Neumann algebras: the classical group crossed product algebras $M \times_{\lambda(e)}{\mathbb{Z}}$ , and certain subalgebras $M_{2}^{\alpha_{e}}$ (M) of operator-valued matricial algebra $M \otimes_{\mathbb{C}} M_{2}({\mathbb{C}})$ . This shows that graph von Neumann algebras identify the key properties of graph groupoids.