# Subshifts, $\lambda$-graph bisystems and $C^*$-algebras

## TL;DR

This paper introduces $mbda$-graph bisystems, establishes their connection to subshifts and $C^*$-algebras, and demonstrates a duality between certain $C^*$-algebras associated with subshifts.

## Contribution

It defines $mbda$-graph bisystems, shows their role in presenting subshifts, and constructs associated $C^*$-algebras with a duality property.

## Key findings

- Any $mbda$-graph bisystem presents a subshift.
- Topological conjugacy of subshifts corresponds to proper strong shift equivalence of symbolic matrix bisystems.
- Associated $C^*$-algebras exhibit a duality, with one isomorphic to a Cuntz-Krieger algebra and the other to a crossed product.

## Abstract

We introduce a notion of $\lambda$-graph bisystem. It consists of a pair $({\frak L}^-, {\frak L}^+)$ of two labeled Bratteli diagrams ${\frak L}^-, {\frak L}^+$ over alphabets $\Sigma^-, \Sigma^+$, respectively, and satisfy certain compatibility condition of their labeling on edges. Its matrix presentation is called a symbolic matrix bisystem. We first show that any $\lambda$-graph bisystem presents subshifts and conversely any subshift is presented by a $\lambda$-graph bisystem, called the canonical $\lambda$-graph bisystem for the subshift. We introduce a notion of properly strong shift equivalence on symbolic matrix bisystems and show that two subshifts are topologically conjugate if and only if their canonical symbolic matrix bisystems are properly strong shift equivalent. A $\lambda$-graph bisystem $({\frak L}^-, {\frak L}^+)$ yields a pair of $C^*$-algebra written ${{\mathcal{O}}_{{\frak L}^-}^+}, {{\mathcal{O}}_{{\frak L}^+}^-}$. We show that the $C^*$-algebras are universal unique $C^*$-algebras subject to certain operator relations among canonical generators encoded by $\lambda$-graph bisystem $({\frak L}^-, {\frak L}^+).$ If a $\lambda$-graph bisystem comes from a $\lambda$-graph system of a finite directed graph, then the associated subshift is the two-sided topological Markov shift $(\Lambda_A, \sigma_A)$ by its transition matrix $A$ of the graph, and the associated $C^*$-algebra ${{\mathcal{O}}_{{\frak L}^-}^+}$ is isomorphic to ${\mathcal{O}}_A,$ whereas the other $C^*$-algebra ${{\mathcal{O}}_{{\frak L}^+}^-}$ is isomorphic to $C(\Lambda_A)\rtimes_{\sigma_A^*}\mathbb{Z}$ of the commutative $C^*$-algebra $C(\Lambda_A)$ on $\Lambda_A$ by the automorphism induced by the homeomorphism of the shift $\sigma_A.$ This phenomena shows a duality between ${\mathcal{O}}_A$ and $C(\Lambda_A)\rtimes_{\sigma_A^*}\mathbb{Z}$.