Primitive ideal space of Higher-rank graph $C^{\ast }$-algebras and decomposability

Abstract

In this paper, we describe primitive ideal space of the $C^{\ast }$-algebra $C^{\ast }(\Lambda )$ 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 $C^{\ast }(\Lambda )$. Furthermore, we study the decomposability of $C^{\ast }(\Lambda )$. We apply the description of primitive ideals to show that if $I$ is a direct summand of $C^{\ast }(\Lambda )$, then it is gauge-invariant and isomorphic to a certain $k$-graph $C^{\ast }$-algebra. So, we may characterize decomposable higher-rank $C^{\ast }$-algebras by giving necessary and sufficient conditions for the underlying $k$-graphs. Moreover, we determine all such $C^{\ast }$-algebras which can be decomposed into a direct sum of finitely many indecomposable $C^{\ast }$-algebras.

The last version

References in corpus (3)

• The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources
• Removing sources from higher-rank graphs
• Simplicity of finitely-aligned k-graph C*-algebras