Topological entropy in C^*-algebras associated with lambda-graph systems

Abstract

A $\lambda$-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. The present author has previously constructed a C^*-algebra $\mathcal{ O}_{\mathfrak L}$ associated with a $\lambda$-graph system $\mathfrak L$ that is a generalization of the Cuntz–Krieger algebras. In this paper, we introduce an entropic quantity for the $\lambda$-graph system $\mathfrak L$, called the volume entropy for $\mathfrak L$ and written as $h_{\rm vol}(\mathfrak L)$. The volume entropy $h_{\rm vol}(\mathfrak L)$ is invariant under shift equivalence of $\lambda$-graph systems, and yields a new topological conjugacy invariant of subshifts. We prove that Voiculescu's non-commutative topological entropy for the canonical completely positive map of the C^*-algebra $\mathcal{O}_{\mathfrak L}$ is the volume entropy $h_{\rm vol}(\mathfrak L)$.