[Paper Review] Higher rank graph C*-algebras

Alex Kumjian, David Pask|ArXiv.org|Jul 5, 2000

Advanced Operator Algebra ResearchMathematics24 references273 citations

TL;DR

This paper introduces higher rank graph C*-algebras as a generalization of graph C*-algebras, defining them via universal C*-algebras generated by partial isometries satisfying factorization and Cuntz-Krieger type relations. The key contribution is establishing an isomorphism between the higher rank graph C*-algebra and the groupoid C*-algebra of its associated path groupoid, enabling the application of groupoid techniques to analyze simplicity, pure infiniteness, and crossed products in this new setting.

ABSTRACT

Building on recent work of Robertson and Steger, we associate a C*-algebra to a combinatorial object which may be thought of as a higher rank graph. This C*-algebra is shown to be isomorphic to that of the associated path groupoid. Sufficient conditions on the higher rank graph are found for the associated C*-algebra to be simple, purely infinite and AF. Results concerning the structure of crossed products by certain natural actions of discrete groups are obtained; a technique for constructing rank 2 graphs from ``commuting'' rank 1 graphs is given.

Motivation & Objective

• To generalize graph C*-algebras to higher rank graphs using a combinatorial category framework with a degree map to N^k.
• To define a universal C*-algebra associated with a higher rank graph using partial isometries satisfying factorization and Cuntz-Krieger relations.
• To establish an isomorphism between the higher rank graph C*-algebra and the C*-algebra of its associated path groupoid.
• To characterize conditions under which the C*-algebra is simple, purely infinite, or AF, using aperiodicity and gauge actions.
• To construct 2-graphs from commuting 1-graphs and analyze their C*-algebras via skew products and crossed products.

Proposed method

• Define a higher rank graph as a small category Λ equipped with a degree map d:Λ→N^k, generalizing directed graphs.
• Construct the path groupoid G_Λ from the infinite path space of Λ, analogous to the path groupoid of a directed graph.
• Prove a gauge-invariant uniqueness theorem that ensures a homomorphism from C*(Λ) is faithful if it is equivariant for the gauge action and nonzero on generators.
• Use the aperiodicity condition on paths in Λ to characterize when the path groupoid is essentially free, enabling a uniqueness theorem analogous to Cuntz-Krieger's.
• Construct skew product k-graphs G×_cΛ from functors c:Λ→G for discrete groups G, and show that the crossed product C*(Λ)⋊_α^cĜ is isomorphic to C*(G×_cΛ).
• Provide a construction of 2-graphs from two 1-graphs A and B with commuting vertex matrices, using a bijection θ:A¹×B¹→B¹×A¹ to define a factorization rule, resulting in a 2-graph A*θB.

Experimental results

Research questions

• RQ1 Under what conditions is the C*-algebra of a higher rank graph simple?
• RQ2 When is the C*-algebra of a higher rank graph purely infinite?
• RQ3 How can the gauge-invariant uniqueness theorem be applied to prove isomorphisms between C*(Λ) and C*(G_Λ)?
• RQ4 What is the structure of crossed products of higher rank graph C*-algebras under group actions, and how do they relate to skew product graphs?
• RQ5 How can 2-graphs be constructed from two commuting 1-graphs, and what are the implications for their C*-algebras?

Key findings

• The C*-algebra C*(Λ) associated with a higher rank graph Λ is isomorphic to the groupoid C*-algebra C*(G_Λ) of its path groupoid, via the gauge-invariant uniqueness theorem.
• The gauge action of T^k on C*(Λ) is implemented by α_t(s_λ) = t^{d(λ)} s_λ, and the crossed product by this action is isomorphic to C*(Z^k ×_d Λ), which is AF.
• If the higher rank graph Λ is aperiodic, then C*(Λ) is simple and purely infinite, generalizing conditions from the Cuntz-Krieger algebra case.
• For a discrete group G acting freely on a k-graph Λ, the crossed product C*(Λ)⋊G is isomorphic to C*(Λ/G)⊗K(ℓ²(G)), generalizing results from graph C*-algebras.
• When two 1-graphs A and B have commuting vertex matrices and are combined via a bijection θ, the resulting 2-graph A*θB has C*-algebra isomorphic to C*(A)⊗C*(B) if θ is the identity, but different if θ is the flip (e.g., O₂*θO₂ ≅ O₂⊗O₂ ≅ O₂, while O₂*ιO₂ ≅ O₂⊗C(T)).
• The construction of 2-graphs from commuting 1-graphs yields a universal model: every 2-graph arises as A*θB for some A, B, and θ, showing the construction is complete up to isomorphism.