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

In this paper, we describe primitive ideal space of the $C^{\ast }$-algebra $C^{\ast }(\mathrm{\Lambda })$ associated to any locally convex row-finite $k$-graph $\mathrm{\Lambda }$. 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 $C^{\ast }(\mathrm{\Lambda })$. Furthermore, we study the decomposability of $C^{\ast }(\mathrm{\Lambda })$. We apply the description of primitive ideals to show that if $I$ is a direct summand of $C^{\ast }(\mathrm{\Lambda })$, 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.