Classification of certain $C^*$-algebras generated by two partitions of unity
Streszczenie
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.
