Kirillov theory for $C^*(G,\Omega)$

TL;DR

This paper extends Kirillov's character theory to the transformation group $C^*$-algebra $C^*(G,\,\Omega)$ for simply connected nilpotent Lie groups, providing a topological parametrization of its primitive ideal space.

Contribution

It constructs a homeomorphism between a quotient space and the primitive ideal space of $C^*(G,\,\Omega)$, generalizing Kirillov's character theory to this setting.

Findings

Established a continuous surjective map to Prim($C^*(G,\,\Omega)$)

Provided a generalized Kirillov character theory for $C^*(G,\,\Omega)$

Described the topology of the primitive ideal space

Abstract

Let $G$ be a simply connected nilpotent Lie group with Lie algebra $\frak g$; let $\frak g^*$ be the dual of $\frak g$. Let $\Omega$ be a locally compact second countable Hausdorff space with a continuous $G$ action, and let $C^*(G,\Omega)$ be the corresponding transformation group $C^*$ algebra. We construct a continuous surjective map $\phi$ from a quotient space, $\frak g^*\times\Omega/\sim$, which is a homeomorphism from $\frak g^*\times\Omega/\sim$ to Prim$(C^*(G,\Omega))$. We also describe a character theory for $C^*(G,\Omega)$ which generalizes Kirillov character theory for $G$.