Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the $C^*$-algebras of linear dynamical systems

Ingrid Beltita, Daniel Beltita

TL;DR

Problem: determine to what extent the C*-algebra $C^*(G_D)$ of a nilpotent linear dynamical system encodes the data of the vector space $V$ and the nilpotent operator $D$ that define the action. Method: apply transformation-groupoid C*-algebras, subquotient analysis via generalized controlled boundary extensions, and limit-point analysis of coadjoint orbits through the Kirillov correspondence. Contributions: (i) verification of RaRo88 for nilpotent groups with a codimension-1 abelian subalgebra; (ii) dimension recovery: dim $V$ and dim $[\mathfrak g,\mathfrak g]$ from $C^*(G)$; (iii) a $C^*$-rigidity result in the low-dimensional regime, showing that if $C^*(G) \cong C^*(G_{D_0})$ then $D_0^2=0$ and the groups are isomorphic. Significance: advances the ability to recover geometric and algebraic data of nilpotent Lie groups from their C*-algebras, contributing to the broader C*-rigidity program.

Abstract

We verify the conjecture on continuous-trace subquotients for $C^*$-algebras of nilpotent linear dynamical systems, where by linear dynamical system we mean a continuous action of the additive group of real numbers by linear maps on a finite-dimensional real vector space. In addition, we show that the dimension of the ambient vector space can be recovered from the corresponding $C^*$-algebra and, if the action is nilpotent of degree two, the corresponding group is $C^*$-rigid within the class of 1-connected nilpotent Lie groups with coadjoint orbits of dimension $\le 2$.

On the $C^*$-algebras of linear dynamical systems

TL;DR

Problem: determine to what extent the C*-algebra of a nilpotent linear dynamical system encodes the data of the vector space and the nilpotent operator that define the action. Method: apply transformation-groupoid C*-algebras, subquotient analysis via generalized controlled boundary extensions, and limit-point analysis of coadjoint orbits through the Kirillov correspondence. Contributions: (i) verification of RaRo88 for nilpotent groups with a codimension-1 abelian subalgebra; (ii) dimension recovery: dim and dim from ; (iii) a -rigidity result in the low-dimensional regime, showing that if then and the groups are isomorphic. Significance: advances the ability to recover geometric and algebraic data of nilpotent Lie groups from their C*-algebras, contributing to the broader C*-rigidity program.

Abstract

We verify the conjecture on continuous-trace subquotients for -algebras of nilpotent linear dynamical systems, where by linear dynamical system we mean a continuous action of the additive group of real numbers by linear maps on a finite-dimensional real vector space. In addition, we show that the dimension of the ambient vector space can be recovered from the corresponding -algebra and, if the action is nilpotent of degree two, the corresponding group is -rigid within the class of 1-connected nilpotent Lie groups with coadjoint orbits of dimension .

Paper Structure

This paper contains 4 sections, 8 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subquotients and generalized controlled boundary extensions
  4. Main results

Key Result

Lemma 2.5

Assume $\lim\limits_{j\to\infty}\xi_j=\xi$ in ${\mathfrak g}^*$ and select arbitrarily ${\mathfrak p}_j\in{\mathfrak S}(\xi_j)$. Then for any cluster point ${\mathfrak p}$ of the sequence $\{{\mathfrak p}_j\}_{j\in{\mathbb N}}$ in ${\rm Gr}({\mathfrak g})$, we have ${\mathfrak p}\in{\mathfrak S}(\xi

Theorems & Definitions (23)

  • Conjecture 1.1: RaRo88
  • Conjecture 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • Example 3.2
  • Definition 3.3
  • ...and 13 more