Primitive ideal space of higher-rank graph C⁎-algebras and decomposability

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