In this paper, we will construct a graph von Neumann algebra M G = M × α G over a fixed von Neumann algebra M, as a crossed product algebra of M and a graph groupoid G induced by a given countable directed graph G, and we will observe 96 I. Cho the amalgamated free probabilistic properties of M G . The construction of such crossed product algebras is motivated by the Exel's crossed product construction. Here, the pair (α w , α −1 w ) is an intertwining analogue of an Exel's interaction, for all w ∈ G. Recently, graph algebras are studied by various operator algebraists and indeed they are very important topics in Operator Algebra because they provides the way to understand operator algebraic objects (for instance, partial isometries and projections) visually. The construction of graph algebras is based on the construction of Cuntz-Kieger algebras. After fixing a Cuntz-Kieger family, we can construct the (topological) operator algebra generated by the family. So, it is natural to restrict our interests to the case when we have "good" graphs satisfying the CuntzKieger relation. The construction of graph von Neumann algebras is free from this restriction. In our construction, we can use any arbitrary countable directed graphs since the construction of graph groupoids is quite canonical. Moreover, by using the (groupoid) crossed product technique, we can study not only a graph algebrawhere v N α (X 1 , X 2 ) means a von Neumann algebra generated by X 1 and X 2 satisfying the relation depending on an action (a partial representation) α of G. Throughout this paper, we concentrate on observing the amalgamated free probabilistic properties of a graph von Neumann algebra M G . This means that we will consider each element of M G as an operator determined by a certain (reduced) word depending on a new algebraic structure G induced by G.

We show that certain representations of graphs by operators on Hilbert space have uses in signal processing and in symbolic dynamics. Our main result is that graphs built on automata have fractal characteristics. We make this precise with the use of Representation Theory and of Spectral Theory of a certain family of Hecke operators. Let G be a directed graph. We begin by building the graph groupoid G induced by G, and representations of G. Our main application is to the groupoids defined from automata. By assigning weights to the edges of a fixed graph G, we give conditions for G to acquire fractal-like properties, and hence we can have fractaloids or G-fractals. Our standing assumption on G is that it is locally finite and connected, and our labeling of G is determined by the "out-degrees of vertices". From our labeling, we arrive at a family of Hecke-type operators whose spectrum is computed. As applications, we are able to build representations by operators on Hilbert spaces (including the Hecke operators); and we further show that automata built on a finite alphabet generate fractaloids. Our Hecke-type operators, or labeling operators, come from an amalgamated free probability construction, and we compute the corresponding amalgamated free moments. We show that the free moments are completely determined by certain scalar-valued functions.

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