Bulletin of the
Korean Mathematical Society BKMS

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).