Higher-rank graph algebras are iterated Cuntz-Pimsner algebras

2017/11/06 by Fletcher, James
#46L05 (Primary) 46L08 #46L45 (Secondary) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1711.01698

Abstract

Given a finitely aligned k-graph Λ, we let Λⁱ denote the (k-1)-graph formed by removing all edges of degree e_i from Λ. We show that the Toeplitz-Cuntz-Krieger algebra of Λ, denoted by TC^*(Λ), may be realised as the Toeplitz algebra of a Hilbert TC^*(Λⁱ)-bimodule. When Λ is locally-convex, we show that the Cuntz-Krieger algebra of Λ, which we denote by C^*(Λ), may be realised as the Cuntz-Pimsner algebra of a Hilbert C^*(Λⁱ)-bimodule. Consequently, TC^*(Λ) and C^*(Λ) may be viewed as iterated Toeplitz and iterated Cuntz-Pimsner algebras over c₀(Λ⁰) respectively.

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