Non‐surjective pullbacks of graph C ^* ‐algebras from non‐injective pushouts of graphs

Abstract

We find a substantial class of pairs of ∗‐homomorphisms between graph C*‐algebras of the form C∗false(Efalse)↪C∗false(Gfalse)↞C∗false(Ffalse) 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 are numerous examples from noncommutative topology, such as quantum complex projective spaces (including the standard Podleś quantum sphere) and 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.

Pullbacks of graph C*-algebras from admissible pushouts of graphs

“…We say that a loop has an exit iff one of its vertices emits an edge not belonging to the loop. We are now ready to prove the main result of this section, which generalizes [7,Theorem 3.4].…”

“…To systematically understand different layers of covariant induction [19], first we observe that the construction of a path algebra defines a covariant functor on the subcategory given by path homomorphisms that are injective on vertices. Then we propose a new monotonicity condition for such path homomorphisms (generalizing [7,Lemma 3.3(1)]), and prove that it defines a further restricted subcategory which yields a covariant functor via the construction of a Cohn path algebra [6]. Finally, we unravel a general regularity condition, vastly generalizing its earlier incarnation [11, Section 2.3] (cf.…”

“…In [7,Theorem 3.4], the authors together with A. Chirvasitu proved a pullback theorem for graph C*-algebras. The theorem was motivated by examples coming from noncommutative topology such as the quantum CW complex structure of the standard Podleś quantum sphere.…”

“…As a key application, we use the aforementioned covariant induction into graph C*-algebras to prove a mixed-pullback theorem (Theorem 5.2) substantially generalizing the mixed-pullback theorem [7,Theorem 3.4]. In particular, we replace the acyclicity assumption in the latter by "all vertex-simple loops have exits" [10,Theorem 2] in the former.…”

Counting paths in directed graphs

“…Graph theory is considered one of the oldest and most accessible branches of combinatorics and has numerous natural connections to other areas of mathematics. In particular, directed graphs, or quivers, are fundamental tools in representation theory [1] as well as in noncommutative geometry [6] and topology [4,3]. In this paper, we focus entirely on the combinatorics of directed graphs, and applications of our results beyond combinatorics are unclear at the moment.…”