The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources

TL;DR

This paper classifies the primitive ideals of Cuntz-Krieger algebras associated with row-finite higher-rank graphs without sources, linking algebraic structure to graph periodicity and maximal tails.

Contribution

It provides a complete description of primitive ideals in these algebras, connecting graph properties with algebraic representations.

Findings

Primitive ideals are indexed by pairs of maximal tails and characters of their periodicity groups.

The algebra is primitive iff the entire vertex set is a maximal tail and the graph is aperiodic.

Maximal tails have abelian periodicity groups of finite rank, at most that of the graph.

Abstract

We catalogue the primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources. Each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph; the primitive ideals in the Cuntz-Krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group. The Cuntz-Krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic.