[Paper Review] The ideal structures of self-similar $k$-graph C*-algebras

Hui Li, Dilian Yang|arXiv (Cornell University)|Jun 25, 2019

Advanced Operator Algebra Research26 references4 citations

TL;DR

This paper establishes a one-to-one correspondence between $G$-hereditary and $G$-saturated subsets of the vertex set $\Lambda^0$ in a self-similar $k$-graph $(G,\Lambda)$ and the gauge-invariant, diagonal-invariant ideals of the associated universal $C^*$-algebra $\mathcal{O}_{G,\Lambda}$. It further characterizes primitive ideals under certain conditions and describes the Jacobson topology for examples like the product of odometers, generalizing structure theorems to the nonunital case and revealing new phenomena in ideal theory.

ABSTRACT

Let $(G, Λ)$ be a self-similar $k$-graph with a possibly infinite vertex set $Λ^0$. We associate a universal C*-algebra $\mathcal{O}_{G,Λ}$ to $(G,Λ)$. The main purpose of this paper is to investigate the ideal structures of $\mathcal{O}_{G,Λ}$. We prove that there exists a one-to-one correspondence between the set of all $G$-hereditary and $G$-saturated subsets of $Λ^0$ and the set of all gauge-invariant and diagonal-invariant ideals of $\mathcal{O}_{G,Λ}$. Under some conditions, we characterize all primitive ideas of $\mathcal{O}_{G,Λ}$. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar $P$-graph C*-algebras in depth.

Motivation & Objective

• To investigate the ideal structure of self-similar $k$-graph $C^*$-algebras $\mathcal{O}_{G,\Lambda}$, which generalize unital $k$-graph algebras and include important examples like Nekrashevych and Exel-Pardo algebras.
• To extend structure theorems from the unital case to the nonunital setting, particularly for $G$-hereditary and $G$-saturated subsets of $\Lambda^0$.
• To characterize primitive ideals of $\mathcal{O}_{G,\Lambda}$ under suitable conditions, especially in relation to periodicity groups and maximal tails.
• To describe the Jacobson topology of the primitive ideal space for concrete examples, including strongly aperiodic $k$-graphs and the product of odometers.

Proposed method

• Introduce and study self-similar $P$-graph $C^*$-algebras as a generalization of self-similar $k$-graphs, which are essential for quotient constructions not closed in the $k$-graph setting.
• Define $G$-hereditary and $G$-saturated subsets of $\Lambda^0$ and prove a one-to-one correspondence with gauge-invariant, diagonal-invariant ideals in $\mathcal{O}_{G,\Lambda}$.
• Use the periodicity group $\mathrm{Per}_{G,\Lambda}$ and its dual to analyze primitive ideals, particularly when $\mathrm{Per}_{G,\Lambda} = \{0\}$.
• Apply the theory to concrete examples: strongly aperiodic $k$-graphs and the product of odometers, where the primitive ideal space is shown to be homeomorphic to $\mathbb{T}^r$ for $r = \mathrm{rank}(\mathrm{Per}_{\Lambda})$.
• Leverage results from [21] on KMS states and tracial states of the periodicity group to connect dynamical properties to ideal structure.
• Use maximal tails $T \in M_\gamma(\Lambda)$ and their associated ideals $I_T = I(\Lambda^0 \setminus T)$ to describe closed sets in the primitive ideal space and prove convergence in the Jacobson topology.

Experimental results

Research questions

• RQ1 What is the precise correspondence between subsets of $\Lambda^0$ and ideals in $\mathcal{O}_{G,\Lambda}$, and how does it extend beyond the unital case?
• RQ2 How do gauge-invariant ideals relate to diagonal-invariant ideals in self-similar $k$-graph $C^*$-algebras, and under what conditions do they coincide?
• RQ3 What conditions ensure that all primitive ideals arise from maximal tails, and how is the primitive ideal space topologized?
• RQ4 How can the Jacobson topology of the primitive ideal space be explicitly described for non-unital examples like the product of odometers?
• RQ5 What role do periodicity groups $\mathrm{Per}_{G,\Lambda}$ play in determining the structure of primitive ideals and the topology of the primitive ideal space?

Key findings

• There exists a one-to-one correspondence between $G$-hereditary and $G$-saturated subsets of $\Lambda^0$ and the gauge-invariant, diagonal-invariant ideals of $\mathcal{O}_{G,\Lambda}$, generalizing results from the unital case.
• A gauge-invariant ideal in $\mathcal{O}_{G,\Lambda}$ is not necessarily diagonal-invariant in general, but this distinction vanishes under mild conditions, such as when the periodicity group is trivial.
• When $\Lambda$ is strongly aperiodic, all ideals of $\mathcal{O}_{G,\Lambda}$ are gauge-invariant and diagonal-invariant, and the primitive ideal space $\mathrm{Prim}(\mathcal{O}_{G,\Lambda})$ is homeomorphic to the set of maximal tails $M_\gamma(\Lambda)$.
• For the product of odometers, $\mathrm{Prim}(\mathcal{O}_{G,\Lambda}) \cong \mathbb{T}^r$ where $r = \mathrm{rank}(\mathrm{Per}_{\Lambda})$, and the Jacobson topology is explicitly described via convergence of maximal tails.
• The closure of a set $Y \subseteq M_\gamma(\Lambda)$ in the primitive ideal space satisfies $T_0 \in \overline{Y}$ if and only if $T_0 \subseteq \bigcup_{T \in Y} T$, providing a topological characterization of closure in terms of maximal tails.