Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I

Abstract

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.

A sampling theory for infinite weighted graphs

“…Both for groupoid actions (see [10,11]) and for networks [20], we study functions on vertices, and on edges. And both approaches make use of combinatorial tools, as well as symbolic dynamics computations in words on edges.…”

Applications of automata and graphs: Labeling operators in Hilbert space. II.

“…Readers can understand the above definition of fractaloids as the graph-groupoid version of the fractal groups (See [1] and [17]). The following theorem provides the graphtheoretical characterization of fractaloids.…”

“…We further address the general case where the vertex set contains more than one element? The answer of this question was provided in [17]. In this paper, we will consider more general case than this.…”

“…Similar to [10] and like [17], we can have a right graph von Neumann algebra M G = C[β(G)] w , as a W * -subalgebra of the operator algebra B(H G ), consisting of all bounded linear operators on H G , where H G is a generalized Fock space, called the graph Hilbert space induced by G. We are interested in the case where G is a fractaloid. Then, similar to the classical case, we can define the Hecke-type operator τ ∈ M G .…”

“…This means that the free moments of τ will contain the operator-theoretical data of τ . In [17], we observed the special cases where the given graph groupoids are fractaloids.…”