Entropy of shifts on higher-rank graph C*-algebras

TL;DR

This paper establishes a relationship between Voiculescu's approximation entropy and topological entropy for higher-rank graph C*-algebras, generalizing previous results for Cuntz-Krieger algebras.

Contribution

It introduces a new entropy formula for higher-rank graph C*-algebras, extending known results to a broader class of algebras.

Findings

ht(Phi^p)=h(T^p) for higher-rank graph C*-algebras

ht(Phi^p)= log r(M_1^{p_1}... M_r^{p_r}) for higher-rank Cuntz-Krieger algebras

Generalizes Boca and Goldstein's result to higher-rank cases

Abstract

Let O_{Lambda} be a higher rank graph C*-algebra of rank r. For every tuple p of non-negative integers there is a canonical completely positive map Phi^p on O_{Lambda} and a subshift T^p on the path space X of the graph. We show that ht(Phi^p)=h(T^p), where ht is Voiculescu's approximation entropy and h the classical topological entropy. For a higher rank Cuntz-Krieger algebra O_M we obtain ht(Phi^p)= log r(M_1^{p_1}M_2^{p_2} ... M_r^{p_r}), r being the spectral radius. This generalises Boca and Goldstein's result for Cuntz-Krieger algebras.