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 number of recent papers have addressed an intriguing interplay between Dis- crete Potential Theory on the one hand and Harmonic Analysis / Spectral Theory on an associated family of fractals on the other, with the Sierpinski gasket serving as a preferred model. This is the first of two papers studying representations of graphs by operators in Hilbert space, and their applications. While graph theory is traditionally considered part of discrete mathematics, in this paper we show that applications of tools from automata and operators on Hilbert spaces yield global results for representations of a class of infinite graphs, as well as spin-off applications. We begin with an outline of the use of automata, and more generally, of finite state models (FSMs) in the processing of numbers, or more importantly in sampling and in quantization of digitized information such as speech signals and digital images. In these models, the finite input states of a particular FSM might be frequency-bands (for example a prescribed pair of high-pass and low-pass digital filters), or a choice of subdivision