Research Projects
Derived quantum graphs (2025 - )
If $\Gamma$ is a directed graph and $\lambda: E_\Gamma \to G$ is a labeling of its edges with elements from a group $G$, then a classical construction of Gross and Tucker yields the derived graph $\Gamma^\lambda$ with vertex and edge set $V_\Gamma \times G$, $E_\Gamma \times G$, respectively. Conversely, a theorem of Gross and Tucker states that every graph $\Gamma$ can be written as derived graph with respect to the group $G$ as soon as there is a free action of $G$ on $\Gamma$. Just one application of this theorem is in the theory of graph C*-algebras, where Kumjian and Pask use it to describe certain graph C*-algebras as crossed product C*-algebras. In an ongoing work with Mariusz Tobolski, we generalize the theory of Gross and Tucker to quantum graphs.
Directed quantum graphs on M2 (2025)
In 2022 D. Gromada and J. Matsuda independently of each other showed that, up to isomorphism, there are exactly four undirected, loopfree, tracial quantum graphs on the matrix algebra M2. Since then, M. Daws and M. Wasilewski have proposed slighly different frameworks for working with non-tracial and possibly non-undirected quantum graphs. In a joint work with Nina Kiefer [1] we pick up the topic of quantum graphs on M2 and classify all quantum graphs on the matrix algebra M2 up to isomorphism - including non-tracial, non-undirected and non-loopfree quantum graphs.
[1] N. Kiefer, B. Schäfer: Complete classification of quantum graphs on M2
Hypergraph C*-algebras and two partitions of unity (2024 - )
This project started with my Master’s thesis supervised by Moritz Weber. Together with M. Trieb and D. Zenner he had introduced hypergraph C*-algebras, a generalization of graph C*-algebras with very different structural properties. The paper [1] investigates which hypergraph C*-algebras are nuclear: It turned out that for a final answer one needs to understand certain C*-algebras generated by two partitions of unity with orthogonality relations by a bipartite graph, aka bipartite graph C*-algebras. By coincidence my other supervisor, Steffen Roch, had investigated Banach algebras generated by two idempotents together with B. Silbermann in the 80’s. Applying their technique, in [2] a very explicit description of hypercube C*-algebras was obtained; in particular these C*-algebras are nuclear. More generally, the paper [3] studies structural properties of bipartite graph C*-algebras. This includes a classification result: Two bipartite graph C*-algebras are isomorphic if and only if their spaces of one- and two-dimensional irreducible representations coincide topologically.
[1] B. Schäfer, M. Weber: Nuclearity of hypergraph C*-algebras
[2] B. Schäfer: Hypercube C*-algebras
[3] B. Schäfer: C*-algebras generated by two partitions of unity