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$\boldsymbol{C}^*$-ALGEBRAS ASSOCIATED WITH TWO-SIDED SUBSHIFTS

Published online by Cambridge University Press:  18 January 2021

KENGO MATSUMOTO*
Affiliation:
Department of Mathematics, Joetsu University of Education, Joetsu 943-8512, Japan
*

Abstract

This 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.

Type
Research Article
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Lisa Orloff Clark

The author was supported by JSPS KAKENHI Grant Nos. 15K04896 and 19K03537.

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