Atomic Representations of Rank 2 Graph Algebras

TL;DR

This paper classifies atomic *-representations of rank 2 graph algebras on a single vertex, decomposing them into irreducible components and describing their minimal *-dilations based on finite groups.

Contribution

It provides a complete classification and decomposition of atomic *-representations of rank 2 graph algebras, introducing a model based on finite groups.

Findings

Complete classification up to unitary equivalence

Decomposition into irreducible atomic representations

Description of building blocks as minimal *-dilations

Abstract

We provide a detailed analysis of atomic *-representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. The building blocks are described as the minimal *-dilations of defect free representations modelled on finite groups of rank 2.