Purely Atomic Representations of Higher-Rank Graph \(\varvec{C}^{\varvec{*}}\)-Algebras

  • Carla Farsi
  • Elizabeth Gillaspy
  • Palle Jorgensen
  • Sooran KangEmail author
  • Judith Packer


We study purely atomic representations of \(C^*\)-algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent and we give necessary and sufficient conditions for a purely atomic representation to be irreducible in terms of the associated projection valued measures. We also investigate the relationship between purely atomic representations, monic representations and permutative representations, and we describe when a purely atomic representation admits a decomposition consisting of permutative representations.


Higher-rank graphs Purely atomic representations Projection-valued measure Permutative representations Irreducible representations 

Mathematics Subject Classification

Primary 46L05 Secondary 46K10 



E.G. was partially supported by the Deutsches Forschungsgemeinschaft via the SFB 878 “Groups, Geometry, and Actions” of the Westfälische-Wilhelms-Universität Münster. S.K.  was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (# NRF-2017R1D1A1B03034697). C.F. and J.P. were partially supported by two individual grants from the Simons Foundation (C.F. #523991; J.P. #316981).


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Colorado at BoulderBoulderUSA
  2. 2.Department of MathematicsUniversity of MontanaMissoulaUSA
  3. 3.Department of Mathematics, 14 MLHUniversity of IowaIowa CityUSA
  4. 4.College of General EducationChung-Ang UniversitySeoulRepublic of Korea

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