The structure of gauge-invariant ideals of labelled graph $C^{\ast }$-algebras

Abstract

In this paper, we consider the gauge-invariant ideal structure of a $C^{\ast }$-algebra $C^{\ast }(E,\mathcal{L},\mathcal{B})$ associated to a set-finite, receiver set-finite and weakly left-resolving labelled space $(E,\mathcal{L},\mathcal{B})$, where $\mathcal{L}$ is a labelling map assigning an alphabet to each edge of the directed graph $E$ with no sinks. Under the assumption that an accommodating set $\mathcal{B}$ is closed under taking relative complement, it is obtained that there is a one to one correspondence between the set of all hereditary saturated subsets of $\mathcal{B}$ and the gauge-invariant ideals of $C^{\ast }(E,\mathcal{L},\mathcal{B})$. For this, we introduce a quotient labelled space $(E,\mathcal{L},[\mathcal{B}{]}_R)$ arising from an equivalence relation ${\sim }_R$ on $\mathcal{B}$ and show the existence of the $C^{\ast }$-algebra $C^{\ast }(E,\mathcal{L},[\mathcal{B}{]}_R)$ generated by a universal representation of $(E,\mathcal{L},[\mathcal{B}{]}_R)$. Also the gauge-invariant uniqueness theorem for $C^{\ast }(E,\mathcal{L},[\mathcal{B}{]}_R)$ is obtained. For simple labelled graph $C^{\ast }$-algebras $C^{\ast }(E,\mathcal{L},\bar{\mathcal{E}})$, where $\bar{\mathcal{E}}$ is the smallest accommodating set containing all the generalized vertices, it is observed that if for each vertex $v$ of $E$, a generalized vertex $[v{]}_l$ is finite for some $l$, then $C^{\ast }(E,\mathcal{L},\bar{\mathcal{E}})$ is simple if and only if $(E,\mathcal{L},\bar{\mathcal{E}})$ is strongly cofinal and disagreeable. This is done by examining the merged labelled graph $(F,{\mathcal{L}}_F)$ of $(E,\mathcal{L})$ and the common properties that $C^{\ast }(E,\mathcal{L},\bar{\mathcal{E}})$ and $C^{\ast }(F,\mathcal{L},\bar{\mathcal{F}})$ share.

References in corpus (2)

• C*-algebras of labelled graphs
• On simple labelled graph $C^{\ast }$-algebras