-ALGEBRAS ASSOCIATED WITH TWO-SIDED SUBSHIFTS

Abstract

AbstractThis paper is a continuation of the paper, Matsumoto [‘Subshifts,$\lambda $-graph bisystems and$C^*$-algebras’,J. Math. Anal. Appl.485 (2020), 123843]. A$\lambda $-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift$\Lambda $, there exists a$\lambda $-graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra${\mathcal {F}}_{\mathcal {L}}$with shift automorphism$\rho _{\mathcal {L}}$from a$\lambda $-graph bisystem$({\mathcal {L}}^-,{\mathcal {L}}^+)$, and define a$C^*$-algebra${\mathcal R}_{\mathcal {L}}$by the crossed product. It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If$\lambda $-graph bisystems come from two-sided subshifts, these$C^*$-algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the$C^*$-algebra${\mathcal R}_{\mathcal {L}}$and the K-theory formulas of the$C^*$-algebras${\mathcal {F}}_{\mathcal {L}}$and${\mathcal R}_{\mathcal {L}}$. The K-group for the AF-algebra${\mathcal {F}}_{\mathcal {L}}$is regarded as a two-sided extension of the dimension group of subshifts.