Journal of Operator Theory

Volume 92, Issue 1, Summer 2024 pp. 25-47.

Relative graphs and pullbacks of relative Toeplitz graph algebras

Authors: Samantha Brooker (1), Jack Spielberg (2)
Author institution:(1) School of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, AZ 85287-1804, U.S.A.
(2) School of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, AZ 85287-1804, U.S.A.

Summary: In this note we generalize a result from a recent paper of Hajac, Reznikoff and Tobolski in \textit{Quantum Dynamics}, Banach Center Publ., vol. 120, Polish Acad. Sci. Inst. Math., Warsaw 2020, pp. 169--178. In that paper they give conditions they call \textit{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 \textit{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.

DOI: http://dx.doi.org/10.7900/jot.2022jul21.2398
Keywords: Toeplitz graph algebra, etale groupoid, relative graph, pullback $C^*$-algebra, gauge action

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