# All classifiable Kirchberg algebras are $C^{\ast}$-algebras of ample groupoids

## TL;DR

This paper proves that all unital Kirchberg algebras in the UCT class can be realized as $C^{ ext{*}}$-algebras of Hausdorff, ample, amenable, and locally contracting groupoids, extending known results to the unital case.

## Contribution

It establishes the realization of unital Kirchberg algebras as groupoid $C^{ ext{*}}$-algebras, building on Spielberg's construction for the non-unital case.

## Key findings

- Unital Kirchberg algebras are groupoid $C^{ ext{*}}$-algebras.
- Extension of groupoid models to the unital case.
- Provides a new construction based on Spielberg's methods.

## Abstract

In this note we show that every Kirchberg algebra in the UCT class is the $C^{\ast}$-algebra of a Hausdorff, ample, amenable and locally contracting groupoid. The non-unital case follows from Spielberg's graph-based models for Kirchberg algebras. Our contribution is the unital case and our proof builds on Spielberg's construction.