Applications of Automata and Graphs: Labeling Operators in Hilbert Space II

TL;DR

This paper explores the use of automata and graph theory in modeling Hilbert space operators, focusing on infinite graphs linked to automaton models, with applications in symbolic dynamics and control theory, and includes a classification theorem for automata representations.

Contribution

It introduces a new framework connecting automata, graphs, and von Neumann algebras, and proves a classification theorem for automata representations in this context.

Findings

Established a classification theorem for automata representations.

Linked automata models to groupoid von Neumann algebras.

Applied the framework to symbolic dynamics and control theory.

Abstract

We introduced a family of infinite graphs directly associated with a class of von Neumann automaton model A_{G}. These are finite state models used in symbolic dynamics: stimuli models and in control theory. In the context of groupoid von Neumann algebras, and an associated fractal group, we prove a classification theorem for representations of automata.