Leavitt vs. $C^*$ pullbacks

TL;DR

This paper demonstrates that specific pullbacks of equivariant *-algebras persist as pullbacks after completion to $C^*$-algebras, unifying various results related to graph algebras and their $C^*$-completions.

Contribution

It establishes that pullbacks of Leavitt path algebras naturally lift to pullbacks of the associated graph $C^*$-algebras, unifying existing literature.

Findings

Pullbacks of certain *-algebras remain pullbacks after $C^*$-completion.

Pullbacks of Leavitt path algebras lift to graph $C^*$-algebras.

Unification of results in graph algebra literature.

Abstract

We show that certain pullbacks of $*$-algebras equivariant with respect to a compact group action remain pullbacks upon completing to $C^*$-algebras. This unifies a number of results in the literature on graph algebras, showing that pullbacks of Leavitt path algebras lift automatically to pullbacks of the corresponding graph $C^*$-algebras.