Relative graphs and pullbacks of relative Toeplitz graph algebras

TL;DR

This paper extends the understanding of when pushouts of relative graphs induce pullback diagrams of their associated $C^*$-algebras, broadening the class of graphs for which this property holds.

Contribution

It generalizes previous results by establishing necessary and sufficient conditions for pullbacks in a larger category of relative Toeplitz graph algebras.

Findings

Provides criteria for pullback diagrams in the category of relative Toeplitz graph algebras.

Extends previous admissibility conditions to a broader class of graphs.

Clarifies the relationship between graph pushouts and $C^*$-algebra pullbacks.

Abstract

In this note we generalize a result from a recent paper of Hajac, Reznikoff and Tobolski (2020). In that paper they give conditions they call admissibility on a pushout diagram in the category of directed graphs implying that the $C^*$-algebras of the graphs form a pullback diagram. We consider a larger category of relative graphs that correspond to relative Toeplitz graph algebras. In this setting we give necessary and sufficient conditions on the pushout to get a pullback of $C^*$-algebras.