# Infinite alphabet edge shift spaces via ultragraphs and their C*-algebras

## TL;DR

This paper introduces a new class of one-sided edge shift spaces associated with ultragraphs, extending finite graph shifts to the infinite case, and explores their topological and algebraic properties, including C*-algebra isomorphisms.

## Contribution

It defines ultragraph-based edge shift spaces for infinite cases, establishing their topological properties and linking conjugate shifts to ultragraph C*-algebra isomorphisms.

## Key findings

- Shift spaces are metrizable with a countable basis of clopen sets.
- For many ultragraphs, basis elements are compact.
- Conjugate shift spaces imply isomorphic ultragraph C*-algebras.

## Abstract

We define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of clopen sets. We show that for a large class of ultragraphs the basis elements of the topology are compact. We examine shift morphisms between these shift spaces, and, for the locally compact case, show that if two (possibly infinite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras are isomorphic. To prove this last result we realize the relevant ultragraph C*-algebras as partial crossed products.