[Paper Review] Ideals in transformation-group $C^*$-algebras

Astrid an Huef, Dana P. Williams|ArXiv.org|Aug 2, 2000

Advanced Operator Algebra Research20 references3 citations

TL;DR

This paper characterizes the largest ideals of specific types—continuous-trace, Fell, liminal, and postliminal—in transformation-group C*-algebras $C_0(X)\rtimes G$, under conditions of abelian or free group actions. It identifies these ideals as crossed products $C_0(Y)\rtimes G$ for explicitly defined $G$-invariant open subsets $Y\subset X$, using dynamical properties such as orbit space $T_0$-separability and wandering sets.

ABSTRACT

We characterize the ideal of continuous-trace elements in a separable transformation-group $C^{*}$-algebra $C_0(X) imes G$. In addition, we identify the largest Fell ideal, the largest liminal ideal and the largest postliminal ideal.

Motivation & Objective

• To characterize the largest continuous-trace ideal in $C_0(X)\rtimes G$ for separable transformation-group C*-algebras.
• To identify the largest Fell, liminal, and postliminal ideals in $C_0(X)\rtimes G$ under abelian or free group actions.
• To express these ideals as $C_0(Y)\rtimes G$ for $G$-invariant open subsets $Y\subset X$, with $Y$ defined via dynamical properties of the action.
• To extend Green’s results on continuous-trace algebras to non-free actions using stability group continuity and $G$-wandering compact sets.
• To unify characterization techniques across different ideal types using duality and orbit space topology, particularly $T_0$-separability.

Proposed method

• Leverages the dual action when $G$ is abelian to ensure ideals are invariant, enabling identification via $G$-invariant open subsets of $X$.
• Applies Phillips’ result that ideals in $C_0(X)\rtimes G$ correspond to $G$-invariant open subsets $Y\subset X$ under EH-regularity conditions.
• Uses Gootman’s characterization: $C_0(X)\rtimes G$ is postliminal iff $X/G$ is $T_0$, and liminal iff all orbits are closed.
• Applies the Glimm–Effros theorem to equate $T_0$-separability of $G\cdot U/G$ with orbit regularity, $G_\delta$-ness, or local closedness.
• Defines key sets $Y$ via neighborhoods $U$ such that $G\cdot U/G$ is $T_0$, and proves maximality via contradiction on orbit closure properties.
• Relies on EH-regularity (ensured by amenable $G$ or $T_0$ orbit space) to guarantee one-to-one correspondence between ideals and $G$-invariant open sets.

Experimental results

Research questions

• RQ1 What is the precise dynamical condition on $X$ that characterizes the largest continuous-trace ideal in $C_0(X)\rtimes G$?
• RQ2 How can the largest postliminal ideal in $C_0(X)\rtimes G$ be described in terms of the orbit space topology when $G$ is abelian or acts freely?
• RQ3 Under what conditions does $C_0(X)\rtimes G$ admit a largest Fell ideal, and how is it realized as a crossed product?
• RQ4 How do properties like orbit regularity, $G_\delta$-ness, or local closedness of orbits relate to the structure of liminal and postliminal ideals?
• RQ5 In what way does the $T_0$-separability of $G\cdot U/G$ determine the maximality of the ideal $C_0(Y)\rtimes G$?

Key findings

• The largest continuous-trace ideal in $C_0(X)\rtimes G$ is $C_0(Y)\rtimes G$, where $Y$ consists of points with a compact wandering neighborhood $N$ such that $q(N)$ is closed and Hausdorff in $X/G$.
• For abelian $G$ or freely acting $G$, the largest postliminal ideal is $C_0(Z)\rtimes G$, where $Z = \{x\in X : \text{some neighborhood } U \text{ of } x \text{ satisfies } G\cdot U/G \text{ is } T_0\}$.
• The largest liminal ideal in $C_0(X)\rtimes G$ is $C_0(Y)\rtimes G$, where $Y$ is the set of points $x\in X$ such that $G\cdot x$ is closed in $X$.
• When $G$ acts freely and $C_0(X)\rtimes G$ is EH-regular, the largest postliminal ideal equals $C_0(Z)\rtimes G$ with $Z$ defined by $T_0$-separability of $G\cdot U/G$.
• The largest Fell ideal in $C_0(X)\rtimes G$ is $C_0(Y)\rtimes G$ where $Y$ is the set of points with neighborhoods $U$ such that $G\cdot U/G$ is $T_0$, provided stability groups vary continuously.
• The characterization of $Y$ as the set of points with $T_0$-separable orbit neighborhoods ensures maximality, proven via contradiction on orbit closure in intermediate sets $Y\cup G\cdot V$.