On outer automorphisms of certain graph $C^{*}$-algebras

TL;DR

This paper constructs specific directed graphs whose associated C*-algebras have K_0 groups isomorphic to any given countable abelian group, and shows how automorphisms of the group lift to automorphisms of the algebra.

Contribution

It provides a method to realize any countable abelian group as the K_0 group of a graph C*-algebra and demonstrates how automorphisms of the group correspond to automorphisms of the algebra.

Findings

Constructed graphs with prescribed K_0 groups

Automorphisms of the abelian group lift to algebra automorphisms

Established a canonical isomorphism between K_0 and the given group

Abstract

Given a countable abelian group $A$, we construct a row finite directed graph $\Gamma(A)$ such that the $K_{0}$-group of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(\Gamma(A))$ is canonically isomorphic to $A$. Moreover, each element of $\textrm {Aut}(A)$ is a lift of an automorphism of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(\Gamma(A))$.