The Primitive Ideal Space of $C(X) \rtimes \mathbb{N}$
Xiaohui Chen, Hui Li
TL;DR
This work addresses the challenge of describing the primitive ideal space and Jacobson topology for a class of topological graph algebras by embedding the problem in the framework of semigroup crossed products. It extends Laca's dilation to injective but not necessarily unital actions, showing Morita equivalence to a group crossed product $C_0(X_\infty)\rtimes_\gamma G$, and, in the unital case, identifies the topological graph algebra $\mathcal{O}(E)$ with $C(X)\rtimes_\alpha \mathbb{N}$ and analyzes its primitive ideals via the quasi-orbit space, yielding a decomposition into periodic and aperiodic components. The main concrete result is an explicit description of Prim$(\mathcal{O}(E))$ in terms of infinite-path data: Prim$(\mathcal{O}(E)) = (\{[e_n]:e_n\in E^\infty_{Per}\}\times \mathbb{T}/\approx) \sqcup \{[e_n]:e_n\in E^\infty_{Aper}\}$. Overall, the paper provides a computable, structured understanding of the primitive ideal space and Jacobson topology for this special class of topological graph algebras, connecting dilation theory, Morita equivalence, and Williams' quasi-orbit techniques.
Abstract
We describe the primitive ideal spaces and the Jacobson topologies of a special class of topological graph algebras.