Simple purely infinite $C^{\ast }$-algebras associated with normal subshifts

Abstract

We will introduce the notion of normal subshift. A subshift ($\Lambda$, $\sigma$) is said to be normal if it satisfies a certain synchronizing property called $\lambda$-synchronizing and is infinite as a set. There are many normal subshifts such as irreducible infinite sofic shifts, Dyck shifts, and $\beta$-shifts whose associated $C^{\ast }$-algebras are simple and purely infinite. Eventual conjugacy of one-sided normal subshifts and topological conjugacy of two-sided normal subshifts are characterized in terms of the associated $C^{\ast }$-algebras and the associated stabilized $C^{\ast }$-algebras with their diagonals and gauge actions, respectively.