arXiv subjects

Björn Schäfer

Publications and source records attributed to Björn Schäfer.

5 recordsLinked to original sources

arXiv2026

Voltage quantum graphs and a Gross-Tucker theorem for quantum graphs

A voltage graph is a finite directed graph whose edges are labeled by elements of a finite group $G$. A classical construction of Gross and Tucker associates to every voltage graph with vertex set $V$ a so-called derived graph with vertex set $V \times G$. We generalize their construction to quantum graphs and finite abelian groups. Remarkably, the construction can produce true quantum graphs starting from a classical voltage graph. In this case the obtained quantum graph is quantum isomorphic to a classical graph. As a main result we also prove a quantum version of the Gross-Tucker theorem which characterizes precisely which graphs can be written as derived graphs of voltage graphs.

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arXiv2025

Hypercube C*-algebras and an application to magic isometries

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.

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arXiv2025

Classification of Certain C*-Algebras Generated by Two Partitions of Unity

We study C*-algebras generated by two partitions of unity subject to orthogonality relations governed by a bipartite graph which we also call "bipartite graph C*-algebras". These algebras generalize at the same time the C*-algebra $C^\ast(p,q)$ generated by two projections and the hypergraph C*-algebras of Trieb, Weber and Zenner. We describe alternative universal generators of bipartite graph C*-algebras and study partitions of unity in generic position associated to a bipartite graph. As a main result, we prove that bipartite graph C*-algebras are completely classified by their one- and two-dimensional irreducible representations which provides a first step towards a classification of the more general hypergraph C*-algebras.

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arXiv2025

Complete Classification of Directed Quantum Graphs on M2

In 2022, Gromada and Matsuda classified undirected quantum graphs on the matrix algebra $M_2$. Later, Wasilweski provided a solid theory of directed quantum graphs which was formerly only established for undirected quantum graphs. Using this framework we extend the results of Matsuda and Gromada, and present a complete classification of directed quantum graphs on $M_2$. Most prominently, we observe that there is a far bigger range of directed quantum graphs than of undirected quantum graphs on $M_2$. Moreover, for quantum graphs on a nontracial quantum set $(M_2, ψ)$ we illustrate the difference between GNS- and KMS-undirected quantum graphs.

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arXiv2024

Nuclearity of Hypergraph C*-Algebras

We partially characterize nuclearity for the recently introduced class of hypergraph C*-algebras using a tailor-made hypergraph minor relation. The latter is generated by certain operations on hypergraphs which resemble the moves on directed graphs used by Eilers, Restorff, Ruiz and Sørensen to classify unital graph C*-algebras. In particular, we obtain a new proof of the fact that every graph C*-algebra associated to a finite graph is nuclear.

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