Gluing topological graph C∗-algebras

In this paper, we introduce regular closed subgraphs of Katsura’s topological graphs and use them to generalize the notion of an adjunction space from topology. Our construction attaches a topological graph onto another via a regular factor map. We prove that under suitable assumptions the [Formula: see text]-algebra of the adjunction graph is a pullback of the [Formula: see text]-algebras of the topological graphs being glued. Our results generalize certain pushout-to-pullback theorems proved in the context of discrete directed graphs. Our theorem applied to homeomorphism [Formula: see text]-algebras recovers a special case of the well-known result stating that pullbacks of [Formula: see text]-[Formula: see text]-algebras induce pullbacks of the respective crossed product [Formula: see text]-algebras. Furthermore, we show that the [Formula: see text]-algebras of odd-dimensional quantum balls of Hong and Szymański (which are known not to be graph [Formula: see text]-algebras) are topological graph [Formula: see text]-algebras and we recover the pullback structure of [Formula: see text]-algebras of odd-dimensional quantum spheres by gluing the topological graphs associated to the [Formula: see text]-algebras of the corresponding odd-dimensional quantum balls.