The Polynomial Endomorphisms of Graph Algebras

TL;DR

This paper characterizes polynomial endomorphisms of graph algebras, linking their automorphism properties to synchronization in coding graphs and dynamics generated by asynchronous transducers.

Contribution

It introduces a new framework connecting polynomial endomorphisms with coding graphs and characterizes when these endomorphisms restrict to automorphisms of the diagonal MASA.

Findings

Provides an if and only if condition for endomorphisms to restrict to automorphisms of the diagonal MASA.

Shows the induced dynamics on the spectrum are generated by an asynchronous transducer.

Establishes a connection between algebraic endomorphisms and symbolic dynamics.

Abstract

We investigate polynomial endomorphisms of graph $C^*$-algebras and Leavitt path algebras. To this end, we define and analyze the coding graph corresponding to each such an endomorphism. We find an if and only if condition for the endomorphism to restrict to an automorphism of the diagonal MASA, which is stated in terms of synchronization of a certain labelling on the coding graph. We show that the dynamics induced this way on the space of infinite paths (the spectrum of the MASA) is generated by an asynchronous transducer.