$C^*$-dynamical systems of solvable Lie groups

TL;DR

This paper introduces a groupoid-based method to analyze the dual spaces of solvable Lie groups via coadjoint actions, revealing structural properties and constructing new examples of Lie groups with specific $C^*$-algebra representations.

Contribution

It develops a novel groupoid approach to the topology of dual spaces of solvable Lie groups and demonstrates how to construct new examples with faithful irreducible representations.

Findings

Coadjoint dynamical system of exponential solvable Lie groups is a piecewise pullback of group bundles.

The approach enables construction of new solvable Lie groups with $C^*$-algebras admitting faithful irreducible representations.

Provides a topological analysis of dual spaces using dynamical systems related to the coadjoint action.

Abstract

In this paper we develop a groupoid approach to some basic topological properties of dual spaces of solvable Lie groups using suitable dynamical systems related to the coadjoint action. One of our main results is that the coadjoint dynamical system of any exponential solvable Lie group is a piecewise pullback of group bundles. Our dynamical system approach to solvable Lie groups also allows us to construct some new examples of connected solvable Lie groups whose $C^*$-algebras admit faithful irreducible representations.