[Paper Review] Periodic higher rank graphs revisited

Dilian Yang|arXiv (Cornell University)|Mar 26, 2014

Graph theory and applications13 references3 citations

TL;DR

This paper introduces a systematic framework for analyzing periodic higher-rank graphs using pullbacks and pushouts of P-graphs, where P is a finitely generated cancellative abelian monoid. It establishes that a periodic P-graph Λ is isomorphic to the pullback of its aperiodic pushout Λ/∼, and shows that its C*-algebra embeds into the tensor product of the C*-algebra of the pushout and C*(PerΛ), leading to the cycline subalgebra M being a maximal abelian subalgebra (MASA).

ABSTRACT

Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph $Λ$ is a natural generalization of a higher rank graph. A pullback of $Λ$ is constructed by pulling it back over a given monoid morphism to $P$, while a pushout of $Λ$ is obtained by modding out its periodicity $\PerΛ$, which is deduced from a natural equivalence relation on $Λ$. One of our main results in this paper shows that, for a class of higher rank graphs $Λ$, $Λ$ 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.

Motivation & Objective

• To develop a general framework for analyzing periodicity in P-graphs, generalizing k-graphs and higher-rank graphs.
• To define and study the pushout of a P-graph by quotienting out its periodicity, resulting in an aperiodic quotient graph.
• To establish that the original P-graph is isomorphic to the pullback of its pushout, thereby linking periodic and aperiodic structures.
• To prove that the cycline C*-subalgebra M of a higher-rank graph is a maximal abelian subalgebra (MASA), resolving a question posed in [2].
• To show that the cycline subalgebra M is isomorphic to the tensor product of the canonical diagonal algebra and C*(PerΛ), and admits a faithful conditional expectation.

Proposed method

• Construct the pullback f*Γ of a Q-graph Γ via a monoid morphism f:P→Q, generalizing the pullback construction from [3].
• Define an equivalence relation ∼ on a P-graph Λ based on shared initial and final degrees, and define PerΛ as the group generated by differences of equivalent paths.
• Construct the pushout Λ/∼ as a quotient graph over the image of P under the quotient map q, resulting in a q(P)-graph.
• Prove that the pushout Λ/∼ is aperiodic under suitable conditions, using factorization and degree comparison arguments.
• Establish an isomorphism Λ ≅ q*(Λ/∼), showing that the original graph is recovered as the pullback of its pushout.
• Use the embedding theorem to show C*(Λ) embeds into C*(Λ/∼) ⊗ C*(PerΛ), leveraging the structure of the cycline subalgebra M.

Experimental results

Research questions

• RQ1 Can the structure of a periodic P-graph be reconstructed from its pushout via pullback?
• RQ2 Under what conditions is the pushout of a P-graph aperiodic?
• RQ3 Is the cycline subalgebra M of a higher-rank graph a maximal abelian subalgebra (MASA) in its C*-algebra?
• RQ4 Can the C*-algebra of a periodic P-graph be embedded into a tensor product of the C*-algebra of its pushout and C*(PerΛ)?
• RQ5 What is the precise structure of the cycline subalgebra M in terms of the canonical diagonal and periodic unitaries?

Key findings

• The original P-graph Λ is isomorphic to the pullback of its pushout Λ/∼, i.e., Λ ≅ q*(Λ/∼), establishing a duality between periodic and aperiodic structures.
• The graph C*-algebra C*(Λ) embeds into the tensor product C*(Λ/∼) ⊗ C*(PerΛ), and this embedding is optimal in general.
• The cycline subalgebra M, generated by equivalent pairs of paths, is isomorphic to D_Λ ⊗ C*(PerΛ), where D_Λ is the canonical diagonal algebra.
• The cycline subalgebra M is a maximal abelian subalgebra (MASA) of C*(Λ), and there exists a faithful conditional expectation from C*(Λ) onto M.
• The commutant of the diagonal algebra D_Λ in C*(Λ) is precisely M, confirming that M is the abelian core of C*(Λ).
• If Λ is finite and cofinal, then C*(Λ/∼) is simple and the center of C*(Λ) is C*(W_h : h ∈ PerΛ), generalizing known results on simplicity and center.