Primitive ideal space of Higher-rank graph $C^*$-algebras and decomposability

TL;DR

This paper characterizes the primitive ideal space and decomposability of higher-rank graph $C^*$-algebras, providing new insights into their structure and conditions for decomposition into indecomposable components.

Contribution

It introduces a detailed description of primitive ideals for $C^*( ext{Lambda})$ using desourcifying methods and characterizes when these algebras are decomposable.

Findings

Primitive ideal space characterized for any locally convex row-finite $k$-graph.

Decomposability conditions for higher-rank $C^*$-algebras established.

All decomposable $C^*$-algebras are characterized as direct sums of indecomposable ones.

Abstract

In this paper, we describe primitive ideal space of the $C^*$-algebra $C^*(\Lambda)$ associated to any locally convex row-finite $k$-graph $\Lambda$. To do this, we will apply the Farthing's desourcifying method on a recent result of Carlsen, Kang, Shotwell, and Sims. We also characterize certain maximal ideals of $C^*(\Lambda)$. Furthermore, we study the decomposability of $C^*(\Lambda)$. We apply the description of primitive ideals to show that if $I$ is a direct summand of $C^*(\Lambda)$, then it is gauge-invariant and isomorphic to a certain $k$-graph $C^*$-algebra. So, we may characterize decomposable higher-rank $C^*$-algebras by giving necessary and sufficient conditions for the underlying $k$-graphs. Moreover, we determine all such $C^*$-algebras which can be decomposed into a direct sum of finitely many indecomposable $C^*$-algebras.