# GCR and CCR Steinberg algebras

## TL;DR

This paper characterizes when the algebra of an ample groupoid over a field is CCR or GCR, providing analogues to $C^*$-algebra results and classifying CCR and GCR Leavitt path algebras.

## Contribution

It offers new characterizations for CCR and GCR properties of ample groupoid algebras and classifies CCR and GCR Leavitt path algebras.

## Key findings

- Characterization of CCR and GCR groupoid algebras
- Analogues to $C^*$-algebra classifications
- Classification of CCR and GCR Leavitt path algebras

## Abstract

Kaplansky introduced the notions of CCR and GCR $C^*$-algebras because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR $C^*$-algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.