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

Part of: Ergodic theory Selfadjoint operator algebras Topological dynamics

Published online by Cambridge University Press:  18 January 2021

KENGO MATSUMOTO
KENGO MATSUMOTO*
Affiliation:
Department of Mathematics, Joetsu University of Education, Joetsu943-8512, Japan
*
e-mail: [email protected]
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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.

Keywords

AF-algebraBratteli diagramC*-algebracrossed productdimension groupK-groupRuelle algebraSmale spacesubshifttopological conjugacy

MSC classification

Primary: 37A55: Relations with the theory of $C^*$-algebras
Secondary: 46L55: Noncommutative dynamical systems 37B10: Symbolic dynamics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 2 , April 2022 , pp. 230 - 263
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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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