Bull. Korean Math. Soc. 2024; 61(1): 217-227
Online first article January 22, 2024 Printed January 31, 2024
https://doi.org/10.4134/BKMS.b230088
Copyright © The Korean Mathematical Society.
Chengjun Hou, Xiangqi Qiang
Yangzhou University; Yangzhou University
Let $G$ be an infinite countable group and $A$ be a finite set. If $ \Sigma \subseteq A^{G}$ is a strongly irreducible subshift of finite type and $\mathcal{G}$ is the local conjugacy equivalence relation on $ \Sigma$. We construct a decreasing sequence $\mathcal{R}$ of unital $C^*$-subalgebras of $C(\Sigma)$ and a sequence of faithful conditional expectations $\mathcal{E}$ defined on $C(\Sigma)$, and obtain a Toeplitz algebra $\mathcal{T}(\mathcal{R},\mathcal{E})$ and a $C^*$-algebra $C^*(\mathcal{R},\mathcal{E})$ for the pair $(\mathcal{R},\mathcal{E})$. We show that $C^*(\mathcal{R},\mathcal{E})$ is $\ast$-isomorphic to the reduced groupoid $C^*$-algebra $C_r^*(\mathcal{G})$.
Keywords: Local conjugacy equivalence relation, Toeplitz algebras, reduced groupoid $C^*$-algebra
MSC numbers: 46L05, 46L85
Supported by: This work was financially supported by the NSF of China (Grant No. 12271469,11971419).
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