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

Abstract 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. [ABSTRACT FROM AUTHOR]