Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs

Abstract

In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we consider free probability on AG.

Keywords

• Directed Graphs
• Graph Groupoids
• Groupoid Dynamical Systems
• 05E15
• 11G15
• 11R04
• 11R09
• 11R47
• 11R56
• 46L10
• 46L40
• 46L53
• 46L54