[Paper Review] The structure of gauge-invariant ideals of labelled graph $C^*$-algebras

Ja A Jeong, Sun Ho Kim+1 more|arXiv (Cornell University)|Feb 21, 2011

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper establishes a one-to-one correspondence between hereditary saturated subsets of an accommodating set and gauge-invariant ideals in labelled graph C*-algebras $C^*(E,\mathcal{L},\mathcal{B})$ under the condition that $\mathcal{B}$ is closed under relative complement. It introduces a quotient labelled space and proves a gauge-invariant uniqueness theorem, leading to a characterization of simplicity: $C^*(E,\mathcal{L},\overline{\mathcal{E}})$ is simple if and only if the labelled space is both strongly cofinal and disagreeable, provided each generalized vertex $[v]_l$ is finite for some $l$. This extends known results for graph and ultragraph C*-algebras to the more general labelled graph setting.

ABSTRACT

In this paper, we consider the gauge-invariant ideal structure of a $C^*$-algebra $C^*(E,\mathcal{L},\mathcal{B})$ 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 $C^*(E,\mathcal{L},\mathcal{B})$. 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 $C^*$-algebra $C^*(E,\mathcal{L},[\mathcal{B}]_R)$ generated by a universal representation of $(E,\mathcal{L},[\mathcal{B}]_R)$. Also the gauge-invariant uniqueness theorem for $C^*(E,\mathcal{L},[\mathcal{B}]_R)$ is obtained. For simple labelled graph $C^*$-algebras $C^*(E,\mathcal{L},\bar{\mathcal{E}})$, 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 $C^*(E,\mathcal{L},\bar{\mathcal{E}})$ 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 $C^*(E,\mathcal{L},\bar{\mathcal{E}})$ and $C^*(F,\mathcal{L},\bar{\mathcal{F}})$ share.

Motivation & Objective

• To determine the structure of gauge-invariant ideals in $C^*(E,\mathcal{L},\mathcal{B})$ for set-finite, receiver set-finite, weakly left-resolving labelled spaces with $\mathcal{B}$ closed under relative complement.
• To establish a one-to-one correspondence between hereditary saturated subsets of $\mathcal{B}$ and gauge-invariant ideals of $C^*(E,\mathcal{L},\mathcal{B})$.
• To introduce a quotient labelled space $(E,\mathcal{L},[\mathcal{B}]_{\textsc{R}})$ and prove the existence of the associated $C^*$-algebra and a gauge-invariant uniqueness theorem.
• To characterize simplicity of $C^*(E,\mathcal{L},\overline{\mathcal{E}})$, where $\overline{\mathcal{E}}$ is the smallest accommodating set closed under relative complement and containing all generalized vertices.
• To show that simplicity of $C^*(E,\mathcal{L},\overline{\mathcal{E}})$ is equivalent to the labelled space being strongly cofinal and disagreeable under the finiteness condition on generalized vertices.

Proposed method

• The paper introduces an equivalence relation $\sim_{\textsc{R}}$ on the accommodating set $\mathcal{B}$, leading to a quotient labelled space $(E,\mathcal{L},[\mathcal{B}]_{\textsc{R}})$.
• It constructs the $C^*$-algebra $C^*(E,\mathcal{L},[\mathcal{B}]_{\textsc{R}})$ via a universal representation of the quotient labelled space.
• A gauge-invariant uniqueness theorem is established for $C^*(E,\mathcal{L},[\mathcal{B}]_{\textsc{R}})$, ensuring that $*$-homomorphisms are injective if they are non-degenerate on the generating projections.
• The paper defines and analyzes the merged labelled space $(F,\mathcal{L}_F)$ of $(E,\mathcal{L})$, which preserves key structural properties such as strong cofinality and disagreeability.
• It proves that strong cofinality and disagreeability are preserved under merging, using the correspondence between labelled paths in $E$ and $F$ and the behavior of ranges under equivalence classes.
• The analysis relies on properties of generalized vertices $[v]_l$, particularly their finiteness, to relate the structure of $C^*(E,\mathcal{L},\overline{\mathcal{E}})$ to that of the merged system.

Experimental results

Research questions

• RQ1 Is there a one-to-one correspondence between hereditary saturated subsets of $\mathcal{B}$ and gauge-invariant ideals in $C^*(E,\mathcal{L},\mathcal{B})$ when $\mathcal{B}$ is closed under relative complement?
• RQ2 How does the quotient labelled space $(E,\mathcal{L},[\mathcal{B}]_{\textsc{R}})$ relate to the ideal structure of $C^*(E,\mathcal{L},\mathcal{B})$?
• RQ3 What conditions on the labelled space $(E,\mathcal{L},\overline{\mathcal{E}})$ ensure the simplicity of $C^*(E,\mathcal{L},\overline{\mathcal{E}})$?
• RQ4 How are strong cofinality and disagreeability preserved under the merging of a labelled graph into its merged version?
• RQ5 Under what conditions is $C^*(E,\mathcal{L},\overline{\mathcal{E}})$ simple, given that each generalized vertex $[v]_l$ is finite for some $l$?

Key findings

• There is a one-to-one correspondence between the set of all hereditary saturated subsets of $\mathcal{B}$ and the gauge-invariant ideals of $C^*(E,\mathcal{L},\mathcal{B})$ when $\mathcal{B}$ is closed under relative complement.
• The quotient labelled space $(E,\mathcal{L},[\mathcal{B}]_{\textsc{R}})$ admits a universal $C^*$-algebra $C^*(E,\mathcal{L},[\mathcal{B}]_{\textsc{R}})$, and a gauge-invariant uniqueness theorem holds for this algebra.
• The merged labelled space $(F,\mathcal{L}_F)$ of $(E,\mathcal{L})$ preserves strong cofinality: $(E,\mathcal{L},{\overline{\mathcal{E}}})$ is strongly cofinal if and only if $(F,\mathcal{L}_F,{\overline{\mathcal{F}}})$ is strongly cofinal.
• The merged labelled space also preserves disagreeability: $(E,\mathcal{L},{\overline{\mathcal{E}}})$ is disagreeable if and only if $(F,\mathcal{L}_F,{\overline{\mathcal{F}}})$ is disagreeable.
• For a set-finite, receiver set-finite, weakly left-resolving labelled space $(E,\mathcal{L},{\overline{\mathcal{E}}})$, the $C^*$-algebra $C^*(E,\mathcal{L},{\overline{\mathcal{E}}})$ is simple if and only if $(E,\mathcal{L},{\overline{\mathcal{E}}})$ is both strongly cofinal and disagreeable, provided each generalized vertex $[v]_l$ is finite for some $l$.