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$\boldsymbol{C}^*$ -ALGEBRAS ASSOCIATED WITH TWO-SIDED SUBSHIFTS.
- Source :
-
Journal of the Australian Mathematical Society . Apr2022, Vol. 112 Issue 2, p230-263. 34p. - Publication Year :
- 2022
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 14467887
- Volume :
- 112
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Journal of the Australian Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 155597094
- Full Text :
- https://doi.org/10.1017/S144678872000049X