Entropy of shifts on topological graph $C^*$-algebras

TL;DR

This paper estimates the entropy of non-commutative shifts on topological graph $C^*$-algebras, comparing them with growth entropies of the graphs, and provides detailed examples involving complex topological structures.

Contribution

It introduces entropy estimates for non-commutative shifts on $C^*$-algebras from topological graphs, linking them to graph growth entropies and illustrating with specific examples.

Findings

Entropy estimates for non-commutative shifts are provided.

Comparison between shift entropies and topological graph growth entropies.

Examples include graphs with vertex and edge spaces as unions of circles.

Abstract

We give entropy estimates for two canonical non commutative shifts on $C^*$-algebras associated to some topological graphs $E=(E^0,E^1,s,r)$, defined using a basis of the corresponding Hilbert bimodule $H(E)$. We compare their entropies with the growth entropies associated directly to the topological graph. We illustrate with some examples of topological graphs considered by Katsura, where the vertex and the edge spaces are a union of unit circles and more detailed computations can be done.