Purely Atomic Representations of Higher-Rank Graph $$\varvec{C}^{\varvec{*}}$$ C ∗ -Algebras

Abstract

We study purely atomic representations of $$C^*$$ -algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent and we give necessary and sufficient conditions for a purely atomic representation to be irreducible in terms of the associated projection valued measures. We also investigate the relationship between purely atomic representations, monic representations and permutative representations, and we describe when a purely atomic representation admits a decomposition consisting of permutative representations.