Graphs of C^*-correspondences and Fell bundles

2008/12/31 by Valentin Deaconu, Deaconu, Valentin, Alex Kumjian +5
Mathematics · Physics and Astronomy · #46L05 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #math.OA #msc:46L05

paper · pdf · doi:10.48550/arxiv.0901.0032

To appear in Indiana Univ. Math. J

openalex publication_date 2008/12/31 · arxiv created 2009/02/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define the notion of a Λ-system of C^*-correspondences associated to a higher-rank graph Λ. Roughly speaking, such a system assigns to each vertex of Λ a C^*-algebra, and to each path in Λ a C^*-correspondence in a way which carries compositions of paths to balanced tensor products of C^*-correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of C^*-correspondences to associate a C^*-algebra to each Λ-system. We then construct a Fell bundle over the path groupoid \Gg_Λ and show that the C^*-algebra of the Λ-system coincides with the reduced cross-sectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature.

Citations

• Generalised morphisms of k-graphs: k-morphs

Related

• A construction of C^*-algebras from C^*-correspondences
• Bundles of C*-correspondences over directed graphs and a theorem of Ionescu
• A Stabilization Theorem for Fell Bundles over groupoids
• Realising the C*-algebra of a higher-rank graph as an Exel crossed product
• Adding tails to C*-correspondences