---
abstract: 'We prove a strong classification result for a certain class of $C^{*}$-algebras with primitive ideal space $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$, where $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$ is the one-point compactification of ${\ensuremath{\mathbb{N}}\xspace}$. This class contains the class of graph $C^{*}$-algebras with primitive ideal space $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$. Along the way, we prove a universal coefficient theorem with ideal-related $K$-theory for $C^{*}$-algebras over $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$ whose $\infty$ fiber has torsion-free $K$-theory.'
address:
- |
    Department of Mathematical Sciences\
    University of Copenhagen\
    Universitetsparken 5\
    DK-2100 Copenhagen, Denmark
- |
    Department of Mathematics\
    University of Hawaii, Hilo\
    200 W. Kawili St.\
    Hilo, Hawaii\
    96720-4091 USA
author:
- James Gabe
- Efren Ruiz
title: 'Classification of tight $C^{*}$-algebras over the one-point compactification of ${\ensuremath{\mathbb{N}}\xspace}$'
---

Introduction
============

Continuous fields of $C^{*}$-algebras appear naturally in the theory of $C^{*}$-algebras since every $C^{*}$-algebra with a Hausdorff primitive ideal space is isomorphic to a continuous field of $C^{*}$-algebras with simple fibers (see [@Fell:Acta61] and [@BlanchardKirchberg:JOT04]). The problem of classifying these $C^{*}$-algebras is an important and classical problem in the theory. In general, these algebras are very far from being locally trivial. In a classical paper [@DixmierDouady:BSMF63], Jacques Dixmier and Adrien Douady classified a certain class of continuous fields of $C^{*}$-algebras over $X$ (continuous trace $C^{*}$-algebras) 