Hypercube C*-algebras and an application to magic isometries
DOI:
https://doi.org/10.2140/mscand.2026.132.335Keywords:
partitions of unity, universal C*-algebras, hypergraph C*-algebras, magic unitariesAbstract
We study C*-algebras generated by two partitions of unity with orthogonality relations governed by hypercubes $Q_n$ for $n \in \mathbb{N} \setminus \{0\}$. These “hypercube C*-algebras” are special cases of bipartite graph C*-algebras which have been investigated by the author in a previous work. We prove that the hypercube C*-algebras $C^{\ast}(Q_n)$ are subhomogeneous and obtain an explicit description of each as an algebra of continuous functions from a standard simplex into a finite-dimensional matrix algebra with suitable boundary conditions. Thus, we generalize Pedersen's description of the universal unital C*-algebra $C^{\ast}(p,q)$ of two projections. We use our results to prove that any $2 {\times} 4$ “magic isometry” matrix can be filled up to a $4 {\times} 4$ “magic unitary” matrix. This answers a question from Banica, Skalski and Sołtan.
References
T. Banica and B. Collins, “Integration over quantum permutation groups”, J. Funct. Anal. 242:2 (2007), 641–657. https://doi.org/10.1016/j.jfa.2006.09.005
T. Banica and B. Collins, “Integration over the Pauli quantum group”, J. Geom. Phys. 58:8 (2008), 942–961. https://doi.org/10.1016/j.geomphys.2008.03.002
T. Banica and S. Moroianu, “On the structure of quantum permutation groups”, Proc. Amer. Math. Soc. 135:1 (2007), 21–29. https://doi.org/10.1090/S0002-9939-06-08464-4
T. Banica and I. Nechita, “Flat matrix models for quantum permutation groups”, Adv. in Appl. Math. 83 (2017), 24–46. https://doi.org/10.1016/j.aam.2016.09.001
T. Banica, J. Bichon, and J.-M. Schlenker, “Representations of quantum permutation algebras”, J. Funct. Anal. 257:9 (2009), 2864–2910. https://doi.org/10.1016/j.jfa.2009.04.013
T. Banica, A. Skalski, and P. Sołtan, “Noncommutative homogeneous spaces: the matrix case”, J. Geom. Phys. 62:6 (2012), 1451–1466. https://doi.org/10.1016/j.geomphys.2012.02.003
B. Blackadar, Operator algebras, Encyc. Math. Sci. 122, Springer, 2006. https://doi.org/10.1007/3-540-28517-2
J. Dixmier, $C^*$-algebras, North-Holland Mathematical Library 15, North-Holland, New York, 1977.
P. R. Halmos, “Two subspaces”, Trans. Amer. Math. Soc. 144 (1969), 381–389. https://doi.org/10.2307/1995288
S. Jung and M. Weber, “Models of quantum permutations”, J. Funct. Anal. 279:2 (2020), art. id. 108516. https://doi.org/10.1016/j.jfa.2020.108516
T. Katsura, M. Ogawa, and A. Takeuchi, “On the magic square $C^*$-algebra of size 4”, Ann. Math. Blaise Pascal 29:1 (2022), 99–148. https://doi.org/10.5802/ambp.408
N. Y. Krupnik, Banach algebras with symbol and singular integral operators, Operator Theory: Advances and Applications 26, Birkhäuser, Basel, 1987. https://doi.org/10.1007/978-3-0348-5463-4
G. K. Pedersen, “Measure theory for $C^*$ algebras, II”, Math. Scand. 22 (1968), 63–74. https://doi.org/10.7146/math.scand.a-10871
I. Raeburn and A. M. Sinclair, “The $C^*$-algebra generated by two projections”, Math. Scand. 65:2 (1989), 278–290. https://doi.org/10.7146/math.scand.a-12283
S. Roch and B. Silbermann, “Algebras generated by idempotents and the symbol calculus for singular integral operators”, Integral Equations Operator Theory 11:3 (1988), 385–419. https://doi.org/10.1007/BF01202079
B. Schäfer, “Classification of certain $C^*$-algebras generated by two partitions of unity”, 2025. https://arxiv.org/abs/2509.01309
B. Schäfer and M. Weber, “Nuclearity of hypergraph $C^*$-algebras”, J. Noncommut. Geom. (online publication November 2025). https://doi.org/10.4171/jncg/635
M. Trieb, M. Weber, and D. Zenner, “Hypergraph $C^*$-algebras”, 2024. https://arxiv.org/abs/2405.09214
S. Wang, “Quantum symmetry groups of finite spaces”, Comm. Math. Phys. 195:1 (1998), 195–211. https://doi.org/10.1007/s002200050385
M. Weber, “Quantum permutation matrices”, Complex Anal. Oper. Theory 17:3 (2023), art. id. 37. https://doi.org/10.1007/s11785-023-01335-x
