# Non-surjective pullbacks of graph C*-algebras from non-injective pushouts of graphs

## TL;DR

This paper identifies a broad class of graph C*-algebra homomorphisms whose pullbacks are AF algebras, providing a method to analyze quantum spaces via graph modifications, with applications to noncommutative topology.

## Contribution

It introduces a new class of pullback constructions for graph C*-algebras that yield AF algebras and extends the results to more general graph C*-algebras involving sinks.

## Key findings

- Pullback C*-algebras are AF in a substantial class of cases.
- The results apply to quantum spaces like quantum spheres and teardrops.
- An extension of the theorem to sink-extended graph C*-algebras is provided.

## Abstract

We find a substantial class of pairs of $*$-homomorphisms between graph C*-algebras of the form $C^*(E)\hookrightarrow C^*(G)\twoheadleftarrow C^*(F)$ whose pullback C*-algebra is an AF graph C*-algebra. Our result can be interpreted as a recipe for determining the quantum space obtained by shrinking a quantum subspace. There is a variety of examples from noncommutative topology, such as quantum complex projective spaces (including the standard Podle\'s quantum sphere) or quantum teardrops, that instantiate the result. Furthermore, to go beyond AF graph C*-algebras, we consider extensions of graphs over sinks and prove an analogous theorem for the thus obtained graph C*-algebras.