Gluing topological graph C*-algebras

TL;DR

This paper generalizes the concept of attaching topological graphs via regular subgraphs, proving that their C*-algebras form pullbacks, and applies this to quantum spheres and balls, extending known results in topological graph C*-algebras.

Contribution

It introduces a new method to glue topological graph C*-algebras using regular subgraphs, generalizing pushout-to-pullback theorems and applying to quantum spaces.

Findings

C*-algebra of the adjunction graph is a pullback of component C*-algebras.

Quantum balls' C*-algebras are topological graph C*-algebras.

Pullback structures of quantum spheres are recovered via gluing topological graphs.

Abstract

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 C*-algebra of the adjunction graph is a pullback of the C*-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 C*-algebras recovers a special case of the well-known result stating that pullbacks of $\mathbb{Z}$-C*-algebras induce pullbacks of the respective crossed product C*-algebras. Furthermore, we show that the C*-algebras of odd-dimensional quantum balls of Hong and Szyma\'nski (which are known not to be graph C*-algebras) are topological graph C*-algebras and we recover the pullback structure of C*-algebras of odd-dimensional quantum spheres by gluing the topological graphs associated to the C*-algebras of the corresponding odd-dimensional quantum balls.