$C^*$-diagonal of inductive limit of $1$-dimensional NCCW complexes

TL;DR

This paper proves the existence of a $C^*$-diagonal in the inductive limit of 1-dimensional NCCW complexes with trivial $K_1$-groups and explores limitations in approximating $^*$-homomorphisms between these complexes.

Contribution

It establishes the existence of a $C^*$-diagonal in inductive limits of 1-dimensional NCCW complexes with trivial $K_1$, and analyzes approximation limitations.

Findings

Existence of $C^*$-diagonal in the inductive limit.

Limitations in approximating $^*$-homomorphisms.

Implications for classification of such complexes.

Abstract

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