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

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Abstract

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.

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Acknowledgements

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. Department of Mathematics, University of Colorado at Boulder, Boulder, CO, 80309-0395, USA

    Carla Farsi & Judith Packer

  2. Department of Mathematics, University of Montana, 32 Campus Drive #0864, Missoula, MT, 59812-0864, USA

    Elizabeth Gillaspy

  3. Department of Mathematics, 14 MLH, University of Iowa, Iowa City, IA, 52242-1419, USA

    Palle Jorgensen

  4. College of General Education, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul, 06974, Republic of Korea

    Sooran Kang

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  1. Carla Farsi
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Correspondence to Sooran Kang.

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Farsi, C., Gillaspy, E., Jorgensen, P. et al. Purely Atomic Representations of Higher-Rank Graph \(\varvec{C}^{\varvec{*}}\)-Algebras. Integr. Equ. Oper. Theory 90, 67 (2018). https://doi.org/10.1007/s00020-018-2493-z

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