$C^{\ast }$-diagonal of inductive limit of $1$-dimensional NCCW complexes

Abstract

This paper establishes the existence of a $C^{\ast }$-diagonal in the inductive limit of 1-dimensional NCCW complexes with trivial $K_1$-groups. It also examines some limitations and implications of approximating ${}^{\ast }$-homomorphisms between two such complexes.