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

TL;DR

This paper investigates the structure of $C^*$-algebras arising from nilpotent linear dynamical systems, confirming a conjecture about their continuous-trace subquotients and exploring their rigidity properties.

Contribution

It verifies the conjecture on continuous-trace subquotients for these $C^*$-algebras and shows how the vector space dimension can be recovered from the algebra, also establishing $C^*$-rigidity in specific cases.

Findings

Confirmed the conjecture on continuous-trace subquotients.

Demonstrated the recoverability of vector space dimension from the $C^*$-algebra.

Established $C^*$-rigidity for nilpotent actions of degree two.

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 $\le 2$.