On the magic square C*-algebra of size 4
Annales mathématiques Blaise Pascal, Volume 29 (2022) no. 1, pp. 99-148.

In this paper, we investigate the structure of the magic square C*-algebra A(4) of size 4. We show that a certain twisted crossed product of A(4) is isomorphic to the homogeneous C*-algebra M 4 (C(P 3 )). Using this result, we show that A(4) is isomorphic to the fixed point algebra of M 4 (C(P 3 )) by a certain action. From this concrete realization of A(4), we compute the K-groups of A(4) and their generators.

Published online:
DOI: 10.5802/ambp.408
Classification: 46L05, 46L55, 46L80
Keywords: C*-algebra, magic square C*-algebra, twisted crossed product, K-theory
Takeshi Katsura 1; Masahito Ogawa 2; Airi Takeuchi 3

1 Department of Mathematics Faculty of Science and Technology Keio University 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522 JAPAN
2 Library & Information center Yokohama City University 22-2 Seto, Kanazawa-ku, Yokohama 236-0027 JAPAN
3 Karlsruhe Institute of Technology Department of Mathematics 76128 Karlsruhe GERMANY
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AMBP_2022__29_1_99_0,
     author = {Takeshi Katsura and Masahito Ogawa and Airi Takeuchi},
     title = {On the magic square {C*-algebra} of size 4},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {99--148},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {29},
     number = {1},
     year = {2022},
     doi = {10.5802/ambp.408},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.408/}
}
TY  - JOUR
AU  - Takeshi Katsura
AU  - Masahito Ogawa
AU  - Airi Takeuchi
TI  - On the magic square C*-algebra of size 4
JO  - Annales mathématiques Blaise Pascal
PY  - 2022
SP  - 99
EP  - 148
VL  - 29
IS  - 1
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.408/
DO  - 10.5802/ambp.408
LA  - en
ID  - AMBP_2022__29_1_99_0
ER  - 
%0 Journal Article
%A Takeshi Katsura
%A Masahito Ogawa
%A Airi Takeuchi
%T On the magic square C*-algebra of size 4
%J Annales mathématiques Blaise Pascal
%D 2022
%P 99-148
%V 29
%N 1
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.408/
%R 10.5802/ambp.408
%G en
%F AMBP_2022__29_1_99_0
Takeshi Katsura; Masahito Ogawa; Airi Takeuchi. On the magic square C*-algebra of size 4. Annales mathématiques Blaise Pascal, Volume 29 (2022) no. 1, pp. 99-148. doi : 10.5802/ambp.408. https://ambp.centre-mersenne.org/articles/10.5802/ambp.408/

[1] Teodor Banica; Julien Bichon Quantum groups acting on 4 points, J. Reine Angew. Math., Volume 626 (2009), pp. 75-114 | DOI | MR | Zbl

[2] Teodor Banica; Benoît Collins Integration over the Pauli quantum group, J. Geom. Phys., Volume 58 (2008) no. 8, pp. 942-961 | DOI | MR | Zbl

[3] Teodor Banica; Sergiu Moroianu On the structure of quantum permutation groups, Proc. Am. Math. Soc., Volume 135 (2007) no. 1, pp. 21-29 | DOI | MR | Zbl

[4] Julien Bichon; Robert Yuncken Quantum subgroups of the compact quantum group SU -1 (3), Bull. Lond. Math. Soc., Volume 46 (2014) no. 2, pp. 315-328 | DOI | MR | Zbl

[5] Nathanial P. Brown; Narutaka Ozawa C * -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, 88, American Mathematical Society, 2008 | Zbl

[6] S. Osugi The homology theory of quotient spaces of the spheres by the action of the finite groups, 2018 (Master thesis, Keio University)

[7] Mikael Rørdam; F. Larsen; Niels Laustsen An introduction to K-theory for C * -algebras, London Mathematical Society Student Texts, 49, Cambridge University Press, 2000

[8] Christian Voigt On the structure of quantum automorphism groups, J. Reine Angew. Math., Volume 732 (2017), pp. 255-273 | DOI | MR | Zbl

[9] Shuzhou Wang Quantum symmetry groups of finite spaces, Commun. Math. Phys., Volume 195 (1998) no. 1, pp. 195-211 | DOI | MR | Zbl

Cited by Sources: