Entropy of shifts on topological graph \(C^\ast\)-algebras (Q1043783)

A topological graph is a quadruple \((E^0,E^1,r,s)\), where \(E^0, E^1\) are locally compact spaces, \(r:E^1 \to E^0\) is a continuous map and \(s:E^1 \to E^0\) is a local homeomorphism. To each such object one can associate a natural Hilbert \(C^*\)-bimodule over the algebra \(C_0(E^0)\) and further via a Pimsner-type construction a \(C^*\)-algebra \(C^*(E)\). The latter is equipped with a canonical shift-type completely positive transformation \(\Phi\). Another approach based on the groupoid theory leads to a \(C^*\)-algebra \(\mathcal{F}_E\) (which is a subalgebra of \(C^*(E)\)) and a different shift-type transformation \(\Psi:\mathcal{F}_E\to \mathcal{F}_E\). Following the theory developed for standard directed graphs, the author defines the loop and block entropy of a topological graph in terms of the growth of cardinality of certain sets of finite paths. Under certain technical assumptions on the topological graph, the loop and block entropies are shown to provide estimates for the noncommutative topological entropies of the maps \(\Psi\) and \(\Phi\). These general results are illustrated by examples of topological graphs related to solenoids, studied earlier in the series of papers by \textit{T.\,Katsura} [Trans.\ Am.\ Math.\ Soc.\ 356, No.\,11, 4287--4322 (2004; Zbl 1049.46039); Int.\ J.\ Math.\ 17, No.\,7, 791--833 (2006; Zbl 1107.46040); Ergodic Theory Dyn.\ Syst.\ 26, No.\,6, 1805--1854 (2006; Zbl 1136.46041); J.\ Funct.\ Anal.\ 254, No.\,5, 1161--1187 (2008; Zbl 1143.46034)]. The examples show, in particular, that it may happen that the topological entropies of \(\Psi\) and \(\Phi\) are different.