Nuclearity of bipartite graph C*-algebras without K_{2,3}

Determine whether every bipartite graph C*-algebra C*(G)—the universal unital C*-algebra generated by projections (p_x)_{x\in U\cup V} satisfying \sum_{u\in U} p_u = 1 = \sum_{v\in V} p_v and p_u p_v = 0 whenever {u,v} is not an edge— is nuclear whenever the underlying bipartite graph G does not contain the complete bipartite graph K_{2,3} as a subgraph.

Background

For bipartite graph C*-algebras C*(G), it is known that the presence of K_{2,3} as a subgraph obstructs nuclearity. The converse—whether the absence of K_{2,3} guarantees nuclearity—has not been established.

The paper addresses this question in the specific case of hypercubes Q_n, which do not contain K_{2,3}, proving that C*(Q_n) is nuclear. This offers affirmative evidence for the subclass of hypercubes and highlights the broader unresolved status for general bipartite graphs without K_{2,3}.