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Hypercube C*-algebras and an application to magic isometries

Björn Schäfer

TL;DR

The paper analyzes hypercube C*-algebras $C^*(Q_n)$, proving $2^{n-1}$-subhomogeneity and that all corner projections are commutative, which enables a rank-one representation theory via admissible edge weightings. It then provides an explicit description of $C^*(Q_n)$ as functions on the standard simplex $oldsymbol{ riangle}_{n-1}$ with values in $M_{2^{n-1}}$ subject to a $\boldsymbol{t}$-block-diagonal boundary condition, unifying and extending Pedersen’s two-projections model. The irreducible representations are classified as subrepresentations of a family $ ho_{\boldsymbol{t}}$, parameterized by $\boldsymbol{t} \in \boldsymbol{ riangle}_{n-1}$, and the results culminate in an explicit, continuous-field realization of the algebras. Finally, these structural insights solve a problem on magic isometries: any $2\times4$ magic isometry can be completed to a $4\times4$ magic unitary by embedding into the quantum permutation algebra $C(S_4^+)$. The work contributes a concrete, operable model for hypercube graph algebras and connects them to quantum permutation theory with concrete combinatorial representations.

Abstract

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.

Hypercube C*-algebras and an application to magic isometries

TL;DR

The paper analyzes hypercube C*-algebras , proving -subhomogeneity and that all corner projections are commutative, which enables a rank-one representation theory via admissible edge weightings. It then provides an explicit description of as functions on the standard simplex with values in subject to a -block-diagonal boundary condition, unifying and extending Pedersen’s two-projections model. The irreducible representations are classified as subrepresentations of a family , parameterized by , and the results culminate in an explicit, continuous-field realization of the algebras. Finally, these structural insights solve a problem on magic isometries: any magic isometry can be completed to a magic unitary by embedding into the quantum permutation algebra . The work contributes a concrete, operable model for hypercube graph algebras and connects them to quantum permutation theory with concrete combinatorial representations.

Abstract

We study C*-algebras generated by two partitions of unity with orthogonality relations governed by hypercubes for . 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 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 of two projections. We use our results to prove that any "magic isometry'' matrix can be filled up to a "magic unitary'' matrix. This answers a question from Banica, Skalski and Sołtan.
Paper Structure (11 sections, 26 theorems, 111 equations, 5 figures)

This paper contains 11 sections, 26 theorems, 111 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgements
  4. Preliminaries
  5. Bipartite graph C*-algebras
  6. Hypercubes
  7. Hypercube C*-algebras are subhomogeneous
  8. Rank-one representations and edge weightings
  9. Irreducible representations of hypercube C*-algebras
  10. Explicit description of hypercube C*-algebras
  11. Application to magic isometries

Key Result

Theorem 1

The $C^\ast$-algebra $C^\ast(Q_n)$ is $2^{n-1}$-subhomogeneous, and one has for all vertices $x$ of $Q_n$ and for every irreducible representation $\rho$ of $C^\ast(Q_n)$ on a Hilbert space $\mathcal{H}$, We call such a representation rank-one.

Figures (5)

  • Figure 1: The hypercubes $Q_1$, $Q_2$ and $Q_3$
  • Figure 2: The hypercubes $Q_1$, $Q_2$ and $Q_3$ with vertices labeled by binary numbers
  • Figure 3: The subgraph of $Q_n$ induced by $x_{i-1}, x_i, x_{i+1}, x_i^\prime$
  • Figure 4: A square in the hypercube $Q_n$ with edges $e, f, g, h$
  • Figure 5: An admissible edge weighting of $Q_2$

Theorems & Definitions (57)

  • Definition 1.1
  • Theorem 1: Theorem \ref{['hyp::thm:subhomogeneous:hypercube_algebra_is_subhomogeneous']}
  • Theorem 2: Theorem \ref{['hyp::thm:hypercube_cstar_algebras_as_continuous_functions']}
  • Definition 1.3
  • Proposition 3: Proposition \ref{['hyp::prop:quantum_sudoku_fill_up']}
  • Definition 2.1
  • Definition 2.2: schafer_classification_2025
  • Proposition 2.3: schafer_classification_2025
  • Definition 2.5
  • Example 2.6
  • ...and 47 more