Hypercube C*-algebras and an application to magic isometries
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909852718071808 |
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| author | Schäfer, Björn |
| author_facet | Schäfer, Björn |
| contents | 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 as 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hypercube C*-algebras and an application to magic isometries Schäfer, Björn Operator Algebras 46L35, 47C15 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 as 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. |
| title | Hypercube C*-algebras and an application to magic isometries |
| topic | Operator Algebras 46L35, 47C15 |
| url | https://arxiv.org/abs/2510.15586 |