The simplicity of the C*-algebras associated to arbitrary labeled spaces

Abstract

We consider the simplicity of the $C^{\ast }$-algebra associated to an arbitrary weakly left-resolving labeled space $(E,\mathcal{ℒ},\mathcal{ℰ})$, where $\mathcal{ℰ}$ is the smallest nondegenerate accommodating set. We classify all gauge-invariant ideals of $C^{\ast }(E,\mathcal{ℒ},\mathcal{ℰ})$ and characterize minimality of $(E,\mathcal{ℒ},\mathcal{ℰ})$ in terms of ideal structure of $C^{\ast }(E,\mathcal{ℒ},\mathcal{ℰ})$. Using these results, we prove that $C^{\ast }(E,\mathcal{ℒ},\mathcal{ℰ})$ is simple if and only if $(E,\mathcal{ℒ},\mathcal{ℰ})$ is strongly cofinal and satisfies condition $(\text{L})$, and for any $A\in \mathcal{ℰ}\setminus \{\varnothing \}$ and $B\in \mathcal{ℰ}$, there is $C\in {\mathcal{ℰ}}_{\text{reg}}$ such that $B\setminus C\in \mathcal{\mathscr{H}}(A)$, and if and only if $(E,\mathcal{ℒ},\mathcal{ℰ})$ is minimal and satisfies condition $(\text{L})$, and if and only if $(E,\mathcal{ℒ},\mathcal{ℰ})$ is minimal and satisfies condition $(\text{K})$.