Periodic higher rank graphs revisited

Abstract

Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph $\Lambda$ is a natural generalization of a higher rank graph. A pullback of $\Lambda$ is constructed by pulling it back over a given monoid morphism to $P$, while a pushout of $\Lambda$ is obtained by modding out its periodicity $\PerΛ$, which is deduced from a natural equivalence relation on $\Lambda$. One of our main results in this paper shows that, for a class of higher rank graphs $\Lambda$, $\Lambda$ is isomorphic to the pullback of its pushout via a natural quotient map, and that its graph C*-algebra can be embedded into the tensor product of the graph C*-algebra of its pushout and $\ca(\PerΛ)$. As a consequence, its cycline C*-algebra generated by the standard generators with equivalent pairs is an abelian core (particularly a MASA). Along the way, we give an in-depth study on periodicity of $P$-graphs.

This replaces the paper arXiv:1403.6848. A mistake in Section 3 is corrected; Section 4 is new, and the title is also changed

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• The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources
• Representations of Higher Rank Graph Algebras
• Type III von Neumann Algebras associated with ${Ø}_{\theta }$
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Cited by in corpus (2)

• KMS states on the C*-algebra of a higher-rank graph and periodicity in the path space
• Cycline subalgebras of $k$-graph C*-algebras