Quantum Cuntz-Krieger algebras
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- by Michael Brannan, Kari Eifler, Christian Voigt and Moritz Weber;
- Trans. Amer. Math. Soc. Ser. B 9 (2022), 782-826
- DOI: https://doi.org/10.1090/btran/88
- Published electronically: October 11, 2022
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Abstract:
Motivated by the theory of Cuntz-Krieger algebras we define and study $C^\ast$-algebras associated to directed quantum graphs. For classical graphs the $C^\ast$-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear.
We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to $KK$-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these $C^\ast$-algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of $KK$-theory.
We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation.
We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.
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Bibliographic Information
- Michael Brannan
- Affiliation: Department of Pure Mathematics and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G0, Canada
- MR Author ID: 887928
- Email: michael.brannan@uwaterloo.ca
- Kari Eifler
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77840
- MR Author ID: 1378155
- Email: keifler.math@gmail.com
- Christian Voigt
- Affiliation: School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom
- MR Author ID: 819355
- ORCID: 0000-0003-3225-5633
- Email: christian.voigt@glasgow.ac.uk
- Moritz Weber
- Affiliation: Saarland University, Faculty of Mathematics, Post Box 151150, D-66041 Saarbrücken, Germany
- MR Author ID: 1008798
- ORCID: 0000-0003-3225-5633
- Email: weber@math.uni-sb.de
- Received by editor(s): February 24, 2021
- Received by editor(s) in revised form: July 13, 2021
- Published electronically: October 11, 2022
- Additional Notes: The first and second authors were partially supported by NSF Grant DMS-2000331. The third and fourth authors were partially supported by SFB-TRR 195 “Symbolic Tools in Mathematics and their Application” at Saarland University. Parts of this project were completed while the authors participated in the March 2019 Thematic Program “New Developments in Free Probability and its Applications” at CRM (Montreal) and the October 2019 Mini-Workshop “Operator Algebraic Quantum Groups” at Mathematisches Forschungs institut Oberwolfach. The authors were supported and provided productive research environments by these institutes.
- © Copyright 2022 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 9 (2022), 782-826
- MSC (2020): Primary 46L55, 46L67, 81P40, 19K35
- DOI: https://doi.org/10.1090/btran/88
- MathSciNet review: 4494623