Ideals in k-graph algebras.

摘要

Higher-rank graph C*-algebras were introduced in 2000 by Kumjian and Pask. They are natural generalizations of directed graph algebras and many of the results concerning higher-rank graphs have come by finding higher-rank analogs of results from the theory of directed graph algebras.;The finitely aligned higher-rank graphs form the largest class of higher-rank graphs with an associated C*-algebra, included amongst them are all row-finite higher-rank graphs and many row-infinite higher-rank graphs. Local periodicity was first identified by Robertson and Sims for row-finite sourceless higher-rank graphs. An appropriate formulation of local periodicity for finitely aligned graphs is introduced and equivalence between the various periodicity conditions is demonstrated. A characterization of the simple finitely aligned higher-rank graph algebras is provided in terms of local periodicity and cofinality.;The primitive ideal structure of higher-rank graph C*-algebras is investigated following a strategy established by an Huef and Raeburn for Cuntz-Krieger algebras and by Bates, Hong, Raeburn, and Szymanski for directed graph algebras. It is shown that primitive ideals naturally correspond with certain subsets of vertices known as maximal tails. It is shown that the gauge-invariant primitive ideals in a higher-rank graph C*-algebra are in bijective correspondence with a sub-collection of maximal tails. This description also provides for an analysis of the hull-kernel topology when every ideal is gauge-invariant. In order to initiate a study of the non-gauge-invariant ideals, a family of irreducible representations is introduced along with a family of commuting unitaries with full joint spectrum. It is shown that the periodic portion of the graph may be separated away from the aperiodic part, which leads to a tensor product decomposition for a large class of periodic higher-rank graph algebras. These results provide the framework for an analysis of the primitive ideal space.