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∗(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.