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 oper- ators 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 correspond- ing von Neumann algebra MG as a groupoid crossed product algebra M ×α G of an arbitrary fixed von Neumann algebra M and the graph groupoid G induced by G, via a graph-representation (or a groupoid action) α. Graph groupoids are well-determined (categorial) groupoids. The graph groupoid G of G has its binary operation, called admissibility. This G has concrete local parts Ge, for all e ∈ E(G). We characterize Me = M ×α Ge of MG, induced by the local parts Ge of G, for all e ∈ E(G). We then characterize all amal- gamated free blocks Me = vN(Me, DG )o fMG. They are chracterized by well-known von Neumann algebras: the classical group crossed product alge- bras M ×λ(e)Z, and certain subalgebras M α e 2 (M ) of operator-valued matricial algebra M ⊗C M2(C). This shows that graph von Neumann algebras identify the key properties of graph groupoids. Mathematics Subject Classification (2000). 47L99.