Hypercube C*-algebras and an application to magic isometries

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 So\l tan.